The Sum-Product Phenomenon and Discrete Geometry

Submitted by Audie Warren

Submitted at Johann Radon Institute for Computational and Applied Mathematics

Supervisor and First Examiner Arne Winterhof

Second Examiner The Sum-Product Misha Rudnev Co-Supervisor Phenomenon and Oliver Roche-Newton Discrete Geometry September 2020

Doctoral Thesis to obtain the academic degree of Doktor der technischen Wissenschaften in the Doctoral Program Technische Wissenschaften

JOHANNES KEPLER UNIVERSITY LINZ Altenbergerstraße 69 4040 Linz, Osterreich¨ www.jku.at DVR 0093696 Declaration

I hereby declare that the thesis submitted is my own unaided work, that I have not used other than the sources indicated, and that all direct and indirect sources are acknowledged as references.

This printed thesis is identical with the electronic version submitted.

Linz, am

Audie Warren

i Abstract

Additive combinatorics is the study of the combinatorial properties of sets of numbers, particularly with respect to the operations of addition and multiplication. This thesis will be primarily concerned with the sum-product phenomenon, which is the principle that a finite subset of a field cannot behave well with respect to both addition and multiplication (unless it is close to being a subfield). Of primary interest are the real numbers and the prime order finite fields.

This thesis uses results from discrete geometry to give improvements to various sum- product type results. Various other results in combinatorial / discrete geometry are proven. It includes joint works with Oliver Roche-Newton, Arne Winterhof, Misha Rudnev, Giorgis Petridis, Mehdi Makhul, and Frank de Zeeuw, and includes results appearing in papers accepted in International Mathematics Research Notices, Proceedings of the American Mathematical So- ciety, Discrete & Computational Geometry, the Moscow Journal of Combinatorics and Number Theory, Finite Fields and Their Applications, and the Electronic Journal of Combinatorics.

Zusammenfassung

Additive Kombinatorik ist die Untersuchung der kombinatorischen Eigenschaften von Mengen von Zahlen, insbesondere im Hinblick auf die Operationen der Addition und Multiplikation. Diese Arbeit befasst sich hauptsächlich mit dem Summenprodukt-Phänomen,bei dem es sich um das Prinzip handelt, dass sich eine endliche Teilmenge eines Körpers sowohl in Bezug auf Addition als auch in Bezug auf Multiplikation nicht gut verhalten kann (es sei denn, es ist nahe daran, ein Teilkörper zu sein). Von primäremInteresse sind die reellen Zahlen und die endlichen Körper erster Ordnung.

Diese Arbeit verwendet Ergebnisse aus diskreter Geometrie, um verschiedene Ergeb- nisse vom Summenprodukttyp zu verbessern. Verschiedene andere Ergebnisse in der kombinatorischen / diskreten Geometrie sind bewiesen. Es enth¨altgemeinsame Arbeiten mit Oliver Roche-Newton, Arne Winterhof, Misha Rudnev, Giorgis Petridis, Mehdi Makhul und Frank de Zeeuw, und Papiere akzeptiert in International Mathematics Research Notices, Proceedings of the American Mathematical Society, Discrete & Computational Geometry, the Moscow Journal of Combinatorics and Number Theory, Finite Fields and Their Applications, und the Electronic Journal of Combinatorics.

Acknowledgements

I would like to thank my supervisors Arne Winterhof and Oliver Roche-Newton for their support and encouragement while writing this thesis. Oliver’s expertise and knowledge on the sum- product phenomenon has been hugely inﬂuential on me, and his ability to explain the main ideas behind a proof is perhaps the most valuable gift a mathematician can have. I would also like to thank all of my co-authors and colleagues, in particular Mehdi Makhul and Misha Rudnev, Mehdi for many discussions on various problems, and Misha for agreeing to be my external referee. This thesis was completed while the author was supported by Austrian Science Fund FWF Project P 30405-N32.

ii Publications

Some parts of this thesis are composed of the following published/accepted works.

• [90] On Products of Shifts in Arbitrary Fields Audie Warren First published in Moscow Journal of Combinatorics and Number Theory, 2019, published by Mathematical Science Publishers.

• [64] New Expander Bounds from Aﬃne Group Energy Oliver Roche-Newton and Audie Warren First published in Discrete & Computational Geometry, 2020, published by Springer Na- ture.

• [53] An Energy Bound in the Aﬃne Group Giorgis Petridis, Oliver Roche-Newton, Misha Rudnev, and Audie Warren First published in International Mathematics Research Notices, 2020, published by Oxford University Press.

• [63] Improved Bounds for Pencils of Lines Oliver Roche-Newton and Audie Warren Accepted in Proceedings of the American Mathematical Society, 2018.

• [45] Constructions for the Elekes - Szab´oand Elekes - R´onyai problems Mehdi Makhul, Oliver Roche-Newton, Frank de Zeeuw, and Audie Warren First published in The Electronic Journal of Combinatorics, 2020.

• [91] Conical Kakeya and Nikodym sets in ﬁnite ﬁelds Audie Warren and Arne Winterhof First published in Finite Fields and Their Applications, 2019, published by Elsevier.

iii Contents

1 Introduction 1 1.1 The Sum-Product Phenomenon ...... 1 1.2 Geometry of Lines ...... 3 1.3 The Elekes-Szab´oand Elekes-R´onyai Problems ...... 5 1.4 The Polynomial Method ...... 6

2 Products of Shifts 7 2.1 Introduction and Main Result ...... 7 2.2 Preliminary Results ...... 8 2.3 Proof of Theorem 7 ...... 12

3 Aﬃne Group Energy and Applications 20 3.1 Preliminaries ...... 20 3.2 Main Results ...... 22 3.2.1 Applications to the Sum-Product Phenomenon ...... 24 3.3 Proofs ...... 25 3.3.1 Proof of Theorems 14 and 15 ...... 25 3.3.2 Proof of Theorem 16 ...... 29 3.3.3 Proof of Corollaries 3, 4, and Theorem 18 ...... 30 3.4 Sum-Product Applications ...... 32 3.4.1 Asymmetric ‘Few Sums Many Products’ Problem ...... 32 3.4.2 The size of AA + A ...... 34 3.4.3 Another Three-Variable Expander ...... 36

4 Pencils of Lines and 4-rich Points 37 4.1 Introduction ...... 37

iv 4.2 Connection with the Sum-Product Problem ...... 38 4.3 Proof of Theorem 5 ...... 39 4.4 Proof of Theorem 26 ...... 41 4.5 Constructions with Arbitrarily many Pencils ...... 44 4.6 Expander Results from Pencils ...... 45

5 Constructions for the Elekes - Szabóand Elekes - Rónyai Problems 50 5.1 Preliminaries ...... 50 5.1.1 The Elekes–SzabóProblem ...... 50 5.1.2 The Elekes–Rónyai Problem ...... 51 5.2 Construction ...... 52 5.3 The Elekes–Rónyai Problem along a Graph ...... 53 5.4 Extensions to more Variables ...... 54 5.4.1 Four Variables ...... 54 5.4.2 More than four Variables ...... 55

6 Kakeya Sets and the Polynomial Method 56 6.1 Preliminaries and Deﬁnitions ...... 56 6.2 Parametrisation of Ellipses ...... 58 6.3 Conical Nikodym Sets ...... 59 6.4 Conical Kakeya Sets ...... 59 6.5 Improvements via the Method of Multiplicities ...... 61 6.5.1 Conical Nikodym Sets ...... 61 6.5.2 Conical Kakeya Sets ...... 63 6.6 Final Remarks ...... 64

7 Open Problems and Further Research 67 7.0.1 The Weak Erd˝os- Szemer´ediConjecture ...... 67 7.0.2 Additive Structure of Squares ...... 68 F2 7.0.3 Collinear Triples in p ...... 70 7.0.4 Paley Graphs and Squares in Diﬀerence Sets ...... 73 7.0.5 Beck’s Theorem over Finite Fields ...... 74

v Notation and Commonly Used Inequalities

Throughout this thesis, the notation , and respectively O(·) and Ω(·) is applied to positive quantities in the usual way. That is, X Y , Y X,X = Ω(Y ), and Y = O(X) are all equivalent and mean that X ≥ cY for some absolute constant c > 0. If both X Y and Y X hold we write X = Θ(Y ).

• The notation X . Y and Y & X both mean that Y X(log X)c for some absolute constant c. If both X . Y and Y . X, we write X ∼ Y .

• For X and Y positive quantities depending on a natural number n, we write X = o(Y ) if X limn→∞ Y = 0.

• The finite field of q elements is denoted by Fq. A finite field of prime order will be denoted by Fp. • The first n natural numbers {1, 2, 3, ..., n} are denoted by [n].

• The symbol F denotes an arbitrary ﬁeld. P(Fn) denotes the projective space over Fn.

We make common use of H¨older’sinequality, which states that for ai, bi complex numbers for i = 1, ..., n, and p, q ∈ (1, ∞) with 1/p + 1/q = 1, we have

n n !1/p n !1/q X X p X q aibi ≤ |ai| |bi| . i=1 i=1 j=1

This inequality is most often applied with p = q = 2, in which case it is named the Cauchy - Schwarz inequality.

vi Chapter 1

Introduction

1.1 The Sum-Product Phenomenon

Additive combinatorics is the study of the structure of sets with respect to addition and multiplication. Two fundamental objects of study in this area are the sum-set and the product-set. Let F be an arbitrary field, and let A ⊂ F be a finite set. We define the sum-set and product-set of A as A + A := {a + b : a, b ∈ A} AA := {ab : a, b ∈ A}. One area of research concerns the sum-product phenomenon, which states that one of these sets should be almost as large as possible, unless A is close to being a subfield. Erd˝osand Szemerédi[31] made this precise in the following conjecture.

Conjecture 1 (Erd˝os- Szemer´edi,1983). For all A ⊂ Z ﬁnite, and all > 0, there exists c = c() > 0 such that |AA| + |A + A| ≥ c|A|2−.

Conjecture 1 is believed to be true over R, where current progress places us at an exponent 4 2 of 3 + 1167 − o(1) due to Rudnev and Stevens [71]. This builds upon the works of Konyagin and Shkredov [40], and Solymosi [82], and the technical improvements of Shakan [75]. Applications of incidence geometry to the sum-product phenomenon began in 1997 with the seminal paper of Elekes [24], where an exponent of 5/4 for Conjecture 1 is proved. The result is a simple but ingenious application of the Szemer´edi- Trotter incidence theorem [84].

Theorem 1 (Szemer´edi- Trotter, 1983). Let P and L be ﬁnite sets of points and lines respectively in R2. Let I(P, L) denote the number of pairs (p, l) ∈ P × L with p ∈ l. Then we have I(P, L) |P |2/3|L|2/3 + |P | + |L|.

Elekes’ argument runs as follows; for a finite set A ⊆ R (we may assume WLOG that 2 0 ∈/ A), define the set of points P := (A + A) × AA ⊆ R , and the set L of lines à,b of the form 2 y = (x − a)b with a, b ∈ A. We have |L| = |A| , and furthermore that for every line à,b ∈ L, the point (a + c, bc) ∈ P lies on à,b for all c ∈ A. Therefore each line in L has at least |A| points lying on it, so that we have |A|3 ≤ I(P, L).

1 Using the Szemer´edi- Trotter theorem to bound from above, we have

|A|3 ≤ I(P, L) |A + A|2/3|AA|2/3|A|4/3 + |A + A||AA| + |A|2.

The ﬁnal error term can be disregarded, the second term leads to a better result than claimed, and therefore we may assume the leading term is dominant, giving

|A + A| + |AA| |A|5/4.

Elekes’ result is seen as the prototypical way of proving a sum-product type result via incidence geometry, and similar applications of incidence theorems to other problems often give non-trivial results which are termed ’threshold bounds’, that is, bounds which follow easily from a single application of an incidence theorem or a simple elementary argument. Conjecture 1 has also been studied over finite fields, particularly fields of prime order. The first result in this direction was proved by Bourgain, Katz, and Tao [12].

δ 1−δ Theorem 2 (Bourgain, Katz, Tao, 2004). Suppose A ⊂ Fp with p < |A| < p for some δ > 0. Then we have 1+(δ) |A + A| + |AA| δ |A| .

Since 2004 there have been advances in incidence bounds over arbitrary (and in particular finite) fields, allowing stronger sum-product results to be proven in this setting. As of writing, the current exponent for the analogue of Conjecture 1 over prime fields is 11/9−o(1) by Rudnev, Shakan, and Shkredov [68]. There are other ways in which the sum-product phenomenon manifests itself. For instance, for A ⊆ R finite, consider the set

A(A + 1) = {a(b + 1) : a, b ∈ A}.

The expectation is that this set is always large, because the translation by 1 in the second product should destroy almost all multiplicative structure. Results have been proven in this direction, the most recent being due to Jones and Roche-Newton [39], who proved that for all ﬁnite A ⊆ R, |A(A + 1)| & |A|24/19. The set A(A + 1) is the simplest example of an expander, that is, a set given by elementary combinations (multiplication and addition) of elements of A, which should be large with respect to |A| regardless of any structure present in A. This is in contrast to the sets A + A and AA, each of which can be made small by a suitable choice of A. Speciﬁcally, if A is an arithmetic progression, e.g. A = {1, 2, ..., n} we have |A + A| = |{2, 3, ..., 2n}| = 2n − 1 = 2|A| − 1 and so the sum-set is small. If we choose A to be a geometric progression we have the same scenario arising for the product set. We expect that the bound

|A(A + 1)| |A|2− holds for all ﬁnite A ⊆ R. A second highly studied expander is the set AA + A. It was conjectured by Balog [4] that |AA + A| ≥ |A|2, however a paper of Roche-Newton, Ruzsa, Shen, and Shkredov [61] gave a construction of a set A such that |AA + A| = o(|A|2). Balog also proved the lower bound

|AA + A| |A|3/2

2 which has subsequently been improved to bounds of the form |AA + A| |A|3/2+δ for some relatively small δ. There are many other similar expanders which have been studied, including the sets A(A + A), (A + A)(A + A) and AA + AA. There are three main new results in this thesis concerning expanders. In Chapter 2 a 1/4 proof is given that for all A ⊆ Fp with |A| < p , we have

|A(A + 1)| & |A|11/9.

In Chapter 3 new results relating to affine group energy are proved, leading to the result that for all A ⊆ R finite, |AA + A| & |A|3/2+1/194. Additionally in Chapter 3 a new four-variable super-quadratic expander is given, that is, a set defined by four variables from a set A ⊆ R, whose size is superquadratic in |A|. Specifically, it is proven that for all A ⊆ R finite, ab − cd 2+1/14 : a, b, c, d ∈ A |A| . a − d In fact, these values correspond to the y-intercepts of the set of lines defined by pairs of points in A × A ⊆ R2.

1.2 Geometry of Lines

This thesis also gives various new results concerning the geometry and combinatorial structure of lines in the plane. Two classical results in this area are the theorems of Beck [7] and Ungar [88], which describe certain properties of the set of lines deﬁned by a point set P , which we denote by L(P ). Speciﬁcally, L(P ) is the set of lines in the plane containing at least two points from P .

Theorem 3 (Beck, 1983). Let P ⊂ R2 be a ﬁnite set of points with at most k points collinear. Then we have |L(P )| |P |(|P | − k).

Theorem 4 (Ungar, 1982). Let P ⊆ R2 be a ﬁnite set of 2N points which do not all lie on a line. Let l∞ denote the line at inﬁnity. Then we have

|L(P ) ∩ l∞| ≥ 2N.

There is an abuse of notation here; L(P )∩l∞ denotes the set of points lying on at least one of the lines in L(P ), and also on l∞. Ungar’s theorem is usually stated in terms of the directions deﬁned by the lines in L(P ), however in order to compare it with a result from Chapter 3 it has been stated projectively. A simple construction showing that Ungar’s theorem is best possible 2 is to consider a regular 2N-gon. Note that l∞ may be changed to any line in P(R ) and the result still holds and is still best possible by considering a projective transformation of the 2N- gon (given that no points of P lie on the line under consideration). This leads to the following 2 question; for two distinct lines l1 and l2 in P(R ), what can we say about |L(P )∩l1|+|L(P )∩l2|? Individually, each term can be made small by the n-gon construction, however we may expect that we cannot have both being small (under certain non-collinearity assumptions of the point

3 set). Such a result is proved in Chapter 3, making use of a new incidence bound of Rudnev and Shkredov [69]. A bound is proved of the form

1+c/14 |L(P ) ∩ l1| + |L(P ) ∩ l2| |P | subject to the condition that no more than O(|P |1−c) points of P lie on a single line for some c > 0. In fact something slightly stronger is proved, see Corollary 3 in Chapter 3 for a precise formulation. The result above makes use of a new bound on affine group energy. Consider the group of (non-vertical and non-horizontal) lines in the plane F2 where F is an arbitrary field, under the group operation of composition. This group is named the affine group Aff(F). Given a finite set of lines L ⊆ Aff(F), one motivating question is to lower bound the size of the set L−1 ◦ L. This has a close connection to the number of solutions to the equation

−1 −1 l1 ◦ l2 = l3 ◦ l4, li ∈ L which is termed the energy of L, denoted E(L). In Chapter 3 a new upper bound on E(L) is given, which then implies the bound

|L|3/2 |L|2 |L−1 ◦ L| + m1/2 M where M is the maximum number of lines in L through a common point in F2, and m is the maximum number of parallel lines in L. Previous related results can be found in [69] and [38, Section 4]. In Chapter 4, a further question relating the geometry of lines and the sum-product phenomenon is considered. A set L of lines is called a pencil if all lines in L pass through a common point (this is considered projectively, hence a set of parallel lines is a pencil). An n-pencil of lines is a pencil of size n, and the centre of a pencil is the common point of the lines. The following question was asked by Rudnev:

Given four n-pencils with non-collinear centres, how many four-rich points do they deﬁne?

A four-rich point is a point lying on a line from each pencil. This problem has a natural connection to the sum-product phenomenon. For a ﬁnite set A ⊆ R, consider the Cartesian product A×A ⊆ R2. The ratio set of A is the set A/A, deﬁned analogously to the sum/product sets. By considering the equation y = rx with r ∈ A/A, it is seen that the size of the ratio set A/A is precisely the number of lines through the origin needed to cover A × A. Indeed, if b (a, b) ∈ A × A, then the line y = a x is the unique line through the origin needed to cover this point, which is of the form y = rx for r ∈ A/A. A similar argument shows that the number of lines of slope −1 needed to cover A × A is precisely |A + A|. Now consider the following four pencils:

•L1 = { lines y = a, a ∈ A}

•L2 = { lines x = a, a ∈ A}

•L3 = { lines y = rx, r ∈ A/A}

•L4 = { lines y = −x + s, s ∈ A + A}.

4 Note that the first two pencils define the Cartesian product A × A. Since the pencils L3 and L4 cover A × A, the four-rich points of these pencils is precisely A × A. The sum-product phenomenon makes us expect that one of L3 and/or L4 must contain many lines. Rudnev’s question asks this in a dual sense; fixing the size of each pencil, how many four rich points can be defined? In Chapter 4 the following result is proved.

Theorem 5. Let P be the set of 4-rich points deﬁned by a set of four non-collinear n-pencils. Then we have |P | = O(n11/6).

This improves a previous result of Chang and Solymosi [16], whose work implied an exponent of 2 − 1/24. The non-collinearity of the centres of the four pencils is necessary in Theorem 5; indeed if A = [n] and the four pencils correspond to the minimal covering sets of vertical lines, horizontal lines, and lines of slope −1 and 1, we obtain four pencils each of size O(n), which deﬁne precisely n2 four-rich points. Indeed, the ﬁrst two pencils are of size precisely n, and the second two pencils have size |A + A| and |A − A| respectively, and for this choice of A we have |A + A|, |A − A| = O(n).

1.3 The Elekes-Szab´oand Elekes-R´onyai Problems

In Chapter 5, we consider two problems regarding polynomials and Cartesian products. The first is called the Elekes-Szabóproblem, see [29] for the originating paper. The problem is whether the zero set of a real polynomial f(x, y, z) can intersect a Cartesian product A×B ×C in many places, where |A| = |B| = |C| = n, and deg(f) = d. In general the intersection of the 2 zero set of f with this Cartesian product can be Ωd(n ), however this should only happen for certain degenerate polynomials. Roughly speaking, degenerate polynomials are ones which can be written as a sum of three univariate functions, for instance a plane can intersect A × B × C in n2 points. Assuming that the polynomial f is non-degenerate, improved bounds on the intersection can be proven, see Chapter 5 for details. Our contribution is to give a bound for this problem from the other side. We give a construction of a non-degenerate polynomial f of constant degree and a Cartesian product A × A × A such that the zero set of f intersects A × A × A in Ω(n3/2) points. It was previously 1+ suggested that the upper bound could be as small as Od(n ), see [20]. Our construction proves that this is not the case, see Theorem 37. The second problem considered is the Elekes-Rónyai problem, which has a similar flavour to the previous problem. The problem is as follows: take a real polynomial f(x, y) = z of degree d, and a Cartesian product A × B with |A| = |B| = n. How large is the image of f on this Cartesian product? Visualised in three dimensions, we take a Cartesian product of points in the x − y plane, and find how many ’heights’ the polynomial f gives on these points. Similarly to the previous problem, we have certain degenerate cases, for instance any plane f(x, y) = C for a constant C gives an image of size one. We again have to define a notion of degeneracy, this time roughly corresponding to f being either a sum or product of univariate polynomials. With such a condition one can prove that the image of f on a Cartesian product must be superlinear in n, see Chapter 5, in particular Theorem 38. We consider this problem along subsets of A × B, which can be seen as bipartite graphs.

5 We give a construction showing that along a subset of a Cartesian product A×A of size Ω(n3/2), a non-degenerate polynomial f can still have an image only of size n, see Theorem 39.

1.4 The Polynomial Method

In Chapter 6, results concerning Kakeya and Nikodym sets in finite fields are proven. A Kakeya Fn set in q is a set of points containing a line in every direction. In 2009, Dvir [21] proved that a n Kakeya set K must have size cnq for some constant cn depending only on n (q is considered the asymptotic parameter in this question). This problem is a finite field analogue of the Kakeya problem in the real setting, see for example [92]. Dvir’s proof followed from an application of the polynomial method (also called Stepanov’s method). This method boils down to ingenious applications of the fact that a degree d polynomial can only have d zeroes. The polynomial method has seen many applications in combinatorics, for example a lower F2 bound on the number of directions defined by a point set in p [85], and recently an improved bound on the clique number of Paley graphs [37], see also the book of Guth [36]. We prove results pertaining to the size of conical Kakeya sets; essentially replacing line with conic in the definition, with the ’direction’ of a conic being its point/s at infinity. Using n the polynomial method, we prove bounds of the form cnq for conical Kakeya sets, applying the method of multiplicities to obtain a constant of 3−n, see Theorem 42. Fn Fn A Nikodym set N in q is a set of points such that for every a ∈ q , there is a line l through a such that l \{a} ⊆ N . The polynomial method can also be used to show that n such sets are large, having size cnq for some constant cn depending only on n. We can define a conical Nikodym set by replacing line with (non-degenerate) conic in the definition. We prove using the method of multiplicities that conical Nikodym sets are also large, having size (3 + o(1))−nqn.

6 Chapter 2

Products of Shifts

In this chapter, we adapt the approach of Rudnev, Shakan, and Shkredov presented in [68] to prove that in an arbitrary ﬁeld F, for all A ⊂ F ﬁnite with |A| < p1/4 if p := Char(F) is positive, we have

|A|11/9 |A|11/9 |A(A + 1)| , |AA| + |(A + 1)(A + 1)| . (log |A|)7/6 (log |A|)7/6

This improves upon the exponent of 6/5 given by an incidence theorem of Stevens and de Zeeuw. The results in this chapter ﬁrst appeared in [90].

2.1 Introduction and Main Result

Conjecture 1 can be considered over arbitrary ﬁelds F. In this setting, we replace the Szemer´edi- Trotter theorem with a point-plane incidence theorem of Rudnev [67], which was used by Stevens and de Zeeuw to derive a point-line incidence theorem [83]. An exponent of 6/5 was proved in 2014 by Roche-Newton, Rudnev, and Shkredov [60]. An application of the Stevens - de Zeeuw theorem gives this exponent of 6/5 for Conjecture 1 over F, so that 6/5 became a threshold to be broken. The 6/5 threshold has recently been broken, see [76], [68], and [17]. The following theorem was proved in [68] by Rudnev, Shakan, and Shkredov, and is the current state of the art bound.

Theorem 6. [68] Let A ⊂ F be a ﬁnite set. If F has positive characteristic p, assume |A| < p18/35. Then we have 11 −o(1) |A + A| + |AA| |A| 9 .

As explained in Chapter 1, another way of considering the sum-product phenomenon is to consider the set A(A + 1), which we would expect to be quadratic in size. This encapsulates the idea that a translation of a set should destroy any multiplicative structure, which is a main theme in sum-product questions. Study of growth of |A(A + 1)| began in [33] by Garaev and Shen, see also [39], [97], and [47]. Current progress for |A(A + 1)| comes from an application of the Stevens - de Zeeuw Theorem, giving the same exponent of 6/5. In this paper we use the multiplicative analogue of ideas in [68] to prove the following theorem.

7 Theorem 7. Let A, B, C, D ⊆ F be ﬁnite with the conditions

|C(A + 1)||A| ≤ |C|3, |C(A + 1)|2 ≤ |A||C|3, |B| ≤ |D|, |A|, |B|, |C|, |D| < p1/4.

Then we have |B|13|A|5|D|3|C| |AB|8|C(A + 1)|2|D(B − 1)|8 . (log |A|)17(log |B|)4

In our applications of this theorem we have |A| = |B| = |C| = |D|, so that the first three conditions are trivially satisfied. The conditions involving p could likely be improved, however for sake of exposition we do not attempt to optimise these. The main proof closely follows [68] (in the multiplicative setting), the central difference being a bound on multiplicative energies in terms of products of shifts. An application of Theorem 7 beats the threshold of 6/5, matching the exponent of 11/9 appearing in Theorem 6. Specifically, we have Corollary 1. Let A ⊆ F be finite, with |A| < p1/4. Then

|A|11/9 |A|11/9 |A(A + 1)| , |AA| + |(A + 1)(A + 1)| . (log |A|)7/6 (log |A|)7/6

Corollary 1 can be seen by applying Theorem 7 with B = A + 1, C = A and D = A + 1 for the ﬁrst result, and B = −A, D = C = A + 1 for the second result.

2.2 Preliminary Results

We require some preliminary theorems. The first is the point-line incidence theorem of Stevens and de Zeeuw. Theorem 8 (Stevens - de Zeeuw, [83]). Let A and B with |B| ≤ |A| be finite subsets of a field F, and let L be a set of lines. Assuming |L||B| p2 and |B||A|2 ≤ |L|3, we have

I(A × B,L) |A|1/2|B|3/4|L|3/4 + |L|.

Note that as |B| ≤ |A|, we have |A|1/2|B|3/4 ≤ |A|3/4|B|1/2, in particular with the same conditions we have the above result with the exponents of A and B swapped. Because of this, the condition |B| ≤ |A| is only needed to specify the second two conditions. We may therefore restate Theorem 8 as follows: Theorem 9. Let A and B be ﬁnite subsets of a ﬁeld F, and let L be a set of lines. Assuming |L| min{|A|, |B|} p2 and |A||B| max{|A|, |B|} ≤ |L|3, we have

I(A × B,L) min{|A|1/2|B|3/4, |A|3/4|B|1/2}|L|3/4 + |L|.

This second formulation will be how we apply Theorem 8. Before stating the next two theorems we require some deﬁnitions. For x ∈ F we deﬁne the representation function n a o r (x) = (a, d) ∈ A × D : = x . A/D d

Note that for all x, rA/D(x) ≤ min{|A|, |D|}. This is seen as ﬁxing one of a, d in the equation a d = x necessarily determines the other. The set A/D in this deﬁnition can be changed to any

8 a R+ other combination of sets, changing the fraction d in the deﬁnition to match. For n ∈ , we deﬁne the n’th moment multiplicative energy of sets A, D ⊆ F as

∗ X n En(A, D) = rA/D(x) . x ∗ ∗ ∗ When n = 2 we shall simply write E (A, D), and when A = D we write En(A) := En(A, A). a ∗ n By considering that we have a = 1 for all a ∈ A, we have the trivial lower bound En(A) ≥ |A| . ∗ when n is in fact a natural number, En(A, D) can be considered as the number of solutions to

a1 a2 an = = ... = ai ∈ A, di ∈ D d1 d2 dn ∗ n giving the trivial upper bound En(A, D) ≤ |A| |D| by fixing a1 to an and then choosing a single di, which necessarily determines all other di. We use Theorem 9 to prove two further results. The first is a bound on the fourth order multiplicative energy relative to products of shifts. Theorem 10. For all finite non-empty A, C, D ⊂ F with |A|2|C(A + 1)| ≤ |D||C|3, |A||C(A + 1)|2 ≤ |D|2|C|3, and |A||C||D|2 p2, we have |C(A + 1)|2|D|3 |C(A + 1)|3|D|2 E∗(A, D) min , log |A|. 4 |C| |C|

The second result is similar, but for the second moment multiplicative energy. Theorem 11. For all ﬁnite and non-empty A, C, D ⊂ F with |A|2|C(A + 1)| ≤ |D||C|3, |A||C(A + 1)|2 ≤ |D|2|C|3, and |A||C||D| min{|C|, |D|} p2, we have |C(A + 1)|3/2|D|3/2 E∗(A, D) log |A|. |C|1/2

The set A + 1 appearing in these theorems can be changed to any translate A + λ for λ 6= 0, by noting that |C(A + 1)| = |C(λA + λ)| and renaming A0 = λA. For our purposes, we will use λ = 1.

Proof of Theorem 10. WLOG we can assume that 0 ∈/ A, C, D. We begin by proving that |C(A + 1)|2|D|3 E∗(A, D) log |A|. 4 |C| Deﬁne the set Sτ := {x ∈ A/D : τ ≤ rA/D(x) < 2τ}. By a dyadic decomposition, there is some τ with

4 ∗ 4 |Sτ |τ E4 (A, D) |Sτ |τ log |A|

Note that τ ≤ min{|A|, |D|}. Take an element t ∈ Sτ . It has τ representations in A/D, so there are τ ways to write t = a/d with a ∈ A, d ∈ D. For all c ∈ C, we have a t = d 1 ac + c − c = d c 1 α = − 1 d c

9 where α = c(a + 1) ∈ C(A + 1). This shows that we have |Sτ |τ|C| incidences between the lines 1 x L = {l : d ∈ D, c ∈ C}, l given by y = − 1 d,c d,c d c 2 and the point set P = C(A + 1) × Sτ . Under the conditions |D||C| min{|Sτ |, |C(A + 1)|} p 3 3 and |Sτ ||C(A + 1)| max{|Sτ |, |C(A + 1)|} ≤ |D| |C| , we have that

1/2 3/4 3/4 3/4 |Sτ |τ|C| ≤ I(P,L) |C(A + 1)| |Sτ | |C| |D| + |D||C|. The conditions are satisﬁed under the assumptions |D||A||C| min{|D|, |C|} p2, |A|2|C(A + 1)| ≤ |D||C|3, and |A||C(A + 1)|2 ≤ |D|2|C|3 . Assuming that the leading term is dominant, we have 4 2 3 |Sτ |τ |C| |C(A + 1)| |D| ∗ E4 (A,D) 4 so that as log |A| |Sτ |τ , we have |C(A + 1)|2|D|3 E∗(A, D) log |A|. 4 |C| We therefore assume the leading term is not dominant. Suppose |D||C| is dominant, so that

1/2 3/4 3/4 3/4 |C(A + 1)| |Sτ | |C| |D| ≤ |D||C|. (2.1) Multiplying by τ 3 and simplifying, we have E∗(A, D)3 |C(A + 1)|2 4 |C(A + 1)|2|S |3τ 12 ≤ |D||C|τ 12 log |A|3 τ |D|1/3|C|1/3τ 4 =⇒ E∗(A, D) log |A|. 4 |C(A + 1)|2/3 The result now follows if |D|1/3|C|1/3τ 4 |C(A + 1)|2|D|3 . |C(A + 1)|2/3 |C| We must therefore prove the result in the case that this is not true; we will prove the result under the assumption |C(A + 1)|2|D|3 |D|1/3|C|1/3τ 4 ≤ |C| |C(A + 1)|2/3 which gives (using τ ≤ |A|)

|D|8|C|4|A|4 ≤ |D|8|C(A + 1)|8 ≤ τ 12|C|4 ≤ |A|12|C|4 so that we have |D| ≤ |A|. We then have (using |C(A + 1)| ≥ |C|1/2|A|1/2)

1/2 3/4 3/4 3/4 1/2 3/4 3/4 1/4 3/4 |D||C| ≥ |C(A+1)| |Sτ | |C| |D| ≥ |C(A+1)| |C| |D| ≥ |A| |C||D| ≥ |D||C| so that the two terms are in fact balanced and the result follows. Secondly, we prove that |C(A + 1)|3|D|2 E∗(A, D) log |A|. 4 |C|

To do this, we swap the roles of D and Sτ from above. We deﬁne the line set and point set by

L = {lt,c : t ∈ Sτ , c ∈ C},P = C(A + 1) × D.

10 Any incidence from the previous point and line set remains an incidence for the new ones, via 1 α 1 α t = d c − 1 ⇐⇒ d = t c − 1 . Under the conditions

2 3 3 |Sτ ||C| min{|D|, |C(A + 1)|} p , |D||C(A + 1)| max{|D|, |C(A + 1)|} ≤ |Sτ | |C| (2.2) we have 3/4 3/4 3/4 1/2 |Sτ |τ|C| ≤ I(P,L) |C(A + 1)| |Sτ | |C| |D| + |Sτ ||C|. ∗ 4 E4 (A,D) If the leading term dominates, the result follows from |Sτ |τ log |A| . Assume the leading term is not dominant, that is,

3 2 |C(A + 1)| |D| ≤ |Sτ ||C|.

Then by using |Sτ | ≤ |A||D| and |A|, |C| ≤ |C(A + 1)| we have

2 2 3 2 |A||C| |D| ≤ |C(A + 1)| |D| ≤ |Sτ ||C| ≤ |A||D||C|

∗ 4 so that |C| = |D| = 1 and the result is trivial by E4 (A, D) ≤ |A||D| ≤ |A|. We now check the conditions (2.2) for using Theorem 8. The first condition in (2.2) is satisfied if |A||C||D|2 p2, which is true under our assumptions. The second condition depends on max{|D|, |C(A + 1)|}, which we assume is |D| (if not the first term in Theorem 10 gives stronger information, which we have already proved). Assuming the second condition does not hold, we have 3 3 2 |Sτ | |C| < |D| |C(A + 1)|. Multiplying by τ 12 and bounding τ ≤ |A|, we get

|A|4|D|2/3|C(A + 1)|1/3 E∗(A, D) log |A|. (2.3) 4 |C|

We may now assume the bound

|C(A + 1)|3|D|2 |A|4|D|2/3|C(A + 1)|1/3 ≤ . (2.4) |C| |C|

Indeed, if we were to have

|A|4|D|2/3|C(A + 1)|1/3 |C(A + 1)|3|D|2 < |C| |C| then we may apply this bound in (2.3) and the result follows. Assuming (2.4), we have

|A|8|D|4 ≤ |C(A + 1)|8|D|4 ≤ |A|12.

So that |D| ≤ |A|. In turn, this implies |A| ≥ |D| ≥ |C(A+1)| ≥ |A|, so that |A| = |C(A+1)| = |D|. Returning to equation (2.3), this gives

|A|4|D|2/3|C(A + 1)|1/3 |C(A + 1)|3|D|2 E∗(A, D) log |A| = log |A| 4 |C| |C| and the result is proved.

11 Proof of Theorem 11. The proof follows similarly to that of Theorem 10. We again deﬁne the lines and points 1 x L = {l : d ∈ D, c ∈ C}, l given by y = − 1 ,P = C(A + 1) × S , d,c d,c d c τ ∗ where in this case the set Sτ is rich with respect to E (A, D), so that

2 ∗ 2 |Sτ |τ E (A, D) |Sτ |τ log |A|.

2 With the conditions |A||C||D| min{|D|, |C|} p and |Sτ ||C(A + 1)| max{|Sτ |, |C(A + 1)|} ≤ |D|3|C|3, (which are satisﬁed under our assumptions) we have by Theorem 9,

1/2 3/4 3/4 3/4 |Sτ |τ|C| ≤ I(P,L) |Sτ | |C(A + 1)| |D| |C| + |D||C|. If the leading term dominates, we have

|C(A + 1)|3/2|D|3/2 |S |τ 2 τ |C|1/2

E∗(A,D) 2 and the result follows from log |A| |Sτ |τ . We therefore assume that the leading term does not dominate, that is, 1/2 3/4 3/4 3/4 |Sτ | |C(A + 1)| |D| |C| ≤ |D||C|. Multiplying through by τ and squaring, we get the bound

|D|1/2|C|1/2τ 2 E∗(A, D) log |A|. (2.5) |C(A + 1)|3/2 In a similar way to before, we may now assume the bound

|D|3/2|C(A + 1)|3/2 |D|1/2|C|1/2τ 2 ≤ (2.6) |C|1/2 |C(A + 1)|3/2 as assuming otherwise yields the result via (2.5). Bound (2.6) then gives

|D||C(A + 1)|3 ≤ |C|τ 2

Bounding τ ≤ |A| and |C||A|2 ≤ |C(A + 1)|3 we have |D| = 1. Similarly, bounding τ 2 ≤ |A||D| and |C(A + 1)|3 ≥ |C|2|A|, we ﬁnd |C| = 1, so that the result is trivial.

2.3 Proof of Theorem 7

WLOG we may assume A, B ⊆ F∗. For some δ > 0, deﬁne a popular set of products as |A||B| P := x ∈ AB : r (x) ≥ . AB |AB|δ Let P c := AB \ P . Note that by writing

|{(a, b) ∈ A × B : ab ∈ P }| + |{(a, b) ∈ A × B : ab ∈ P c}| = |A||B| and noting that |A||B| |A||B| |{(a, b) ∈ A × B : ab ∈ P c}| < |P c| ≤ |AB|δ δ

12 we have 1 |{(a, b) ∈ A × B : ab ∈ P }| ≥ 1 − |A||B|. δ We also deﬁne a popular subset of A with respect to P , as 2 A0 := a ∈ A : |{b ∈ B : ab ∈ P }| ≥ |B| . 3 We have X X 1 |{(a, b) ∈ A×B : ab ∈ P }| = |{b : ab ∈ P }|+ |{b : ab ∈ P }| ≥ 1 − |A||B| (2.7) δ a∈A0 a∈A\A0

Suppose that |A \ A0| = c|A| for some c ≥ 0, so that |A0| = (1 − c)|A|. Noting that X X 2c |{b : ab ∈ P }| ≤ (1 − c)|A||B|, |{b : ab ∈ P }| ≤ |A||B|, 3 a∈A0 a∈A\A0 we have by (2.7) 2c 1 3 (1 − c)|A||B| + |A||B| ≥ (1 − )|A||B| =⇒ c ≤ , 3 δ δ

0 3 so that |A | ≥ 1 − δ |A|. We use a multiplicative version of Lemma 3.1 in [68].

Lemma 1. For all ﬁnite A ⊂ F, there exists A1 ⊆ A with |A1| |A|, such that

∗ 0 ∗ E4/3(A1) E4/3(A1)

Proof. We give an algorithm which shows such a subset exists, as otherwise we have a contradiction. We recursively deﬁne

0 Ai = Ai−1,A0 = A, i ≤ log |A|

0 0 where Ai is deﬁned relative to Ai. Using the same arguments as above, we have |Ai| ≥ 3 1 − δ |Ai|. We shall set δ = log |A|. We have the chain of inequalities 3 3 i |A | = |A0 | ≥ 1 − |A | ≥ ... ≥ 1 − |A|. i i−1 log |A| i−1 log |A| Note that assuming |A| ≥ 16 (if this is not true then the result is trivial), we have

3 i 3 log |A| 14 1 − ≥ 1 − ≥ log |A| log |A| 4

3 z since the function 1 − z is increasing for z > 3. We now have 14 |A | ≥ |A| |A| i 4 at all steps i. We assume that at all steps, we have

∗ E (Ai) E∗ (A0 ) < 4/3 4/3 i 4

13 ∗ 0 ∗ as otherwise we have E4/3(Ai) E4/3(Ai) and we are done. After log |A| steps, we have a set Ak with ∗ ∗ ∗ E (k) E (Ak−1) E (A) |A | |A|,E∗ (A0 ) < 4/3 < 4/3 < ... < 4/3 . k 4/3 k 4 16 4log |A| But then we have ∗ ∗ 0 log |A| 4/3+2 10/3 E4/3(A) > E4/3(Ak)4 |A| = |A| which is a contradiction. Therefore at some step we have an Ai satisfying the lemma.

We now return to the proof of Theorem 7, with δ = log |A| applied in the definition of P . ∗ 0 ∗ We apply Lemma 1 to A to find a large subset A1 ⊂ A with E4/3(A1) E4/3(A1), |A1| |A|. Noting that proving the result for A1 implies it for A, we shall rename A1 as A for simplicity. We use a dyadic decomposition to find a set Q ⊂ A0/A0 such that

4/3 ∗ 0 4/3 |Q|∆ E4/3(A ) |Q|∆ log |A| for some ∆ > 0. We will bound the size of the set n a o N = (a, a0, b, b0) ∈ (A0)2 × B2 : ∈ Q, ab, ab0, a0b, a0b0 ∈ P . a0 0 0 a By summing over all a, a ∈ A with a0 ∈ Q, we have X |N| = |{b ∈ B : ab, a0b ∈ P }|2 a,a0∈A0 :a/a0∈Q

0 2 and we see that as for all a ∈ A , |{b ∈ B : ab ∈ P }| ≥ 3 |B|, by considering the intersection of {b ∈ B : ab ∈ P } and {b ∈ B : a0b ∈ P } we have that for all a, a0 ∈ A0, |{b ∈ B : ab, a0b ∈ 1 0 0 P }| ≥ 3 |B|. Using that elements q ∈ Q have at least ∆ representations in A /A , we have 1 2 |N| ≥ 9 |B| |Q|∆. We now find an upper bound on |N|. Define an equivalence relation on A2 × B2 via d d0 (a, a0, b, b0) ∼ (c, c0, d, d0) ⇐⇒ ∃ λ s.t. a = λc, a0 = λc0, b = , b0 = . λ λ Note that the conditions a ∈ Q, ab, a0b, ab0, a0b0 ∈ P (2.8) a0 are invariant in the class (i.e. if one class element satisfies these conditions, then they all do), as λ cancels in each condition. Let X denote the set of equivalence classes [a, a0, b, b0] where the conditions (2.8) are satisfied. We can bound |N| by the sum of the size of each equivalence class [a, a0, b, b0] in X; X |N| ≤ |[a, a0, b, b0]| . X By the Cauchy-Schwarz inequality and completing the sum over all equivalence classes, we have

X 2 |Q|2∆2|B|4 |N|2 ≤ |X| |[a, a0, b, b0]| . (2.9) [a,a0,b,b0] We must now bound the two quantities on the right hand side of this equation. We ﬁrst claim that X 0 0 2 X 2 2 |[a, a , b, b ]| ≤ rA/A(x) rB/B(x) . (2.10) [a,a0,b,b0] x

14 To see this, note that the left hand side of (2.10) counts pairs of elements of equivalence classes. Take any two elements (a, a0, b, b0), (c, c0, d, d0) ∈ A2 × B2 from the same equivalence 0 0 0 b b0 class. By definition, we may write (c, c , d, d ) = (λa, λa , λ , λ ). As 0 ∈/ A, B, the 8-tuple (a, a0, b, b0, c, c0, d, d0) satisfies c c0 b b0 λ = = = = a a0 d d0 2 2 for some λ ∈ R, and thus corresponds to a contribution to the quantity rA/A(λ) rB/B(λ) , and P 2 2 thus also corresponds to a contribution to the sum x rA/A(x) rB/B(x) . We also see that different pairs from equivalence classes necessarily give different 8-tuples, and so the claim is proved. We use Cauchy-Schwarz on the right hand side of equation (2.10) to bound it by a product of fourth energies.

X 2 2 ∗ 1/2 ∗ 1/2 rA/A(x) rB/B(x) ≤ E4 (A) E4 (B) . x We use Theorem 10 to bound these energies. We bound via |C(A + 1)|2|A|3 |D(B − 1)|2|B|3 E∗(A) log |A|,E∗(B) log |B| 4 |C| 4 |D| with conditions |C(A + 1)||A| ≤ |C|3, |C(A + 1)|2 ≤ |A||C|3, |A|3|C| p2 |D(B − 1)||B| ≤ |D|3, |D(B − 1)|2 ≤ |B||D|3, |B|3|D| p2 which are all satisfied under our assumptions. Returning to equation (2.9), we now have |C(A + 1)||A|3/2|D(B − 1)||B|3/2 |Q|2∆2|B|4 |X| (log |A| log |B|)1/2. (2.11) |C|1/2|D|1/2 We now bound |X|, the number of equivalence classes where the conditions (2.8) are satisfied. Note that any (a, a0, b, b0) belonging to an equivalence class in X maps to a solution of the equation s u w = = (2.12) t v a 0 0 0 0 with w ∈ Q, s, t, u, v ∈ P , by taking w = a0 , s = ab, t = a b, u = ab , v = a b . Note that taking two solutions (a, a0, b, b0) and (c, c0, d, d0) that are not from the same equivalence class necessarily gives us two different solutions to equation (2.12) via the map above. Therefore we may bound |X| by the number of solutions to (2.12). n s uo |X| ≤ (w, s, t, u, v) ∈ Q × P 4 : w = = t v n s u o = (s, t, u, v) ∈ P 4 : = ∈ Q . t v The popularity of P allows us to bound this by 4 4 |AB| (log |A|) 4 4 a1b1 a3b3 |X| ≤ 4 4 (a1, a2, a3, a4, b1, b2, b3, b4) ∈ A × B : = ∈ Q . |A| |B| a2b2 a4b4 BA 0 0 We dyadically pigeonhole the set A in relation to the number of solutions to r/a = r /a ∈ Q 0 BA 0 BA with r, r ∈ A , a, a ∈ A to find popular subsets R1,R2 ⊆ A in terms of these solutions. We have

4 4 2 log |A| |AB| (log |A|) X X 2 3 x a3b3 AB |X| ≤ 4 4 r (x) (a3, a4, b1, b3, b4) ∈ A × B : = ∈ Q . |A| |B| A b1 a4a4 i=1 AB x∈ A i i+1 2 ≤r AB (x)<2 A

15 AB We use the pigeonhole principle to give us ∆1 > 0 and R1 ⊆ A such that 4 5 |AB| (log |A|) 2 3 r1 a3b3 |X| ∆1 4 4 (r1, a3, a4, b2, b3, b4) ∈ R1 × A × B : = ∈ Q . |A| |B| b2 a4b4

0 AB We perform a similar dyadic decomposition to get ∆1 > 0 and R2 ⊆ A such that 4 6 0 |AB| (log |A|) 2 r1 r2 |X| ∆1∆1 4 4 (r1, r2, b2, b4) ∈ R1 × R2 × B : = ∈ Q . |A| |B| b2 b4 These decompositions now allow us to bound via fourth energies, as follows.

4 6 0 |AB| (log |A|) 2 r1 r2 |X| ∆1∆1 4 4 (r1, r2, b2, b4) ∈ R1 × R2 × B : = ∈ Q |A| |B| b2 b4 |AB|4(log |A|)6 X = ∆ ∆0 r (q)r (q) 1 1 |A|4|B|4 R1/B R2/B q∈Q !1/2 !1/2 |AB|4(log |A|)6 X X ≤ ∆ ∆0 r (q)2 r (q)2 1 1 |A|4|B|4 R1/B R2/B q∈Q q∈Q |AB|4(log |A|)6 ≤ ∆ ∆0 |Q|1/2 E∗(B,R )1/4E∗(B,R )1/4 (2.13) 1 1 |A|4|B|4 4 1 4 2 where the third and fourth lines follow from applications of the Cauchy-Schwarz inequality. 2 2 We will now show that given |B||D||Ri| p and |B| ≤ |D| (which are true under our assumptions), we have |D(B − 1)|3|R |2 E∗(B,R ) i log |B|. (2.14) 4 i |D| Firstly, with the additional conditions

2 3 2 2 3 |B| |D(B − 1)| ≤ |Ri||D| , |B||D(B − 1)| ≤ |Ri| |D| (2.15) we may bound these fourth energies by Theorem 10 to get (2.14). We can therefore assume one of these conditions does not hold.

2 3 Firstly, suppose that |B| |D(B − 1)| > |Ri||D| . We will use the trivial bound

∗ 4 E4 (B,Ri) ≤ |Ri| |B|. Note that it would be enough to prove |D(B − 1)|3|R |2 E∗(B,R ) ≤ i 4 i |D| which would follow from |D(B − 1)|3|R |2 |R |4|B| ≤ i (2.16) i |D| 2 3 2 which is true if and only if |Ri| |B||D| ≤ |D(B−1)| . Using our assumed bound |B| |D(B−1)| > 3 |Ri||D| , we know that |B|5|D(B − 1)|2 |R |2|B||D| < . i |D|5 By the assumption |B| ≤ |D|, we have |B|5|D(B − 1)|2 |R |2|B||D| < ≤ |D(B − 1)|3 i |D|5

16 and so by (2.16), the bound on the fourth energy holds. Now assume the second condition from (2.15) does not hold, that is, |B||D(B − 1)|2 > 2 3 |Ri| |D| . Again, we use the trivial bound

∗ 4 E4 (B,Ri) ≤ |Ri| |B|. We have |D(B − 1)|3|R |2 |R |4|B| ≤ i ⇐⇒ |R |2|B||D| ≤ |D(B − 1)|3 i |D| i 2 3 so that it is enough to prove |Ri| |B||D| ≤ |D(B − 1)| , as before. Using the assumption 2 2 3 |B||D(B − 1)| > |Ri| |D| , we have the information that |B|2|D(B − 1)|2 |R |2|B||D| < i |D|2 and it follows from our assumption |B| ≤ |D| that |B|2|D(B − 1)|2 ≤ |D(B − 1)|3. |D|2

2 3 Therefore we have |Ri| |B||D| < |D(B − 1)| and so the bound on the fourth energy holds. Returning to equation (2.13), we use (2.14) to bound |X| as |AB|4(log |A|)6 |X| ∆ ∆0 |Q|1/2 E∗(B,R )1/4E∗(B,R )1/4 1 1 |A|4|B|4 4 1 4 2 |AB|4|D(B − 1)|3/2 ∆ ∆0 |R |1/2|R |1/2|Q|1/2 (log |A|)6(log |B|)1/2 (2.17) 1 1 1 2 |A|4|B|4|D|1/2 As |R |∆ ≤ P r (x), the product |R |1/2|R |1/2∆ ∆0 can be bounded by i i x∈Ri BA/A 1 2 1 1 !1/2 1/2 1/2 0 X 2 X 2 |R1| |R2| ∆1∆1 ≤ r BA (x) r BA (x) A A x∈R1 x∈R2

0 where it is important to note that r BA (x) gives a triple (b, a, a ). For i = 1, 2, we have A X 2 4 2 b1a1 b2a3 r BA (x) ≤ (a1, a2, a3, a4, b1, b2) ∈ A × B : = . A a2 a4 x∈Ri

Following a similar dyadic decomposition as before, we ﬁnd a pair of subsets S1,S2 ⊆ A/A with 0 respect to these solutions, and some ∆2, ∆2 > 0 with

X 2 0 2 2 r BA (x) ∆2∆2(log |A|) (s1, s2, b1, b2) ∈ S1 × S2 × B : s1b1 = s2b2 A x∈Ri 0 2 X ≤ ∆2∆2(log |A|) rS1B(x)rS2B(x) x 0 2 ∗ 1/2 ∗ 1/2 ≤ ∆2∆2(log |A|) E (B,S1) E (B,S2) , where the third inequality is given by the Cauchy-Schwarz inequality. We will use a similar 2 argument as above to prove that with the two conditions |B||D||Si| min{|D|, |Si|} p and |B| ≤ |D| (which are satisﬁed under our assumptions), we have |S |3/2|D(B − 1)|3/2 E∗(B,S ) i log |B|. (2.18) i |D|1/2

17 Under the extra conditions

2 3 2 2 3 |B| |D(B − 1)| ≤ |Si||D| , |B||D(B − 1)| ≤ |Si| |D| (2.19) we can bound this energy by Theorem 11 to get (2.18). We therefore assume the ﬁrst condition 2 3 from (2.19) does not hold, that is, |B| |D(B − 1)| > |Si||D| . We bound the energy via the trivial estimate

∗ 2 E (B,Si) ≤ |B||Si| . It is now enough to show that

|S |3/2|D(B − 1)|3/2 |B||S |2 ≤ i which is true iﬀ |B||D|1/2|S |1/2 ≤ |D(B − 1)|3/2. i |D|1/2 i

2 3 Using our assumption |B| |D(B − 1)| > |Si||D| , we have that |B|2|D(B − 1)|1/2 |B||D|1/2|S |1/2 < . i |D| Our assumption that |B| ≤ |D| then gives

|B|2|D(B − 1)|1/2 ≤ |B||D(B − 1)|1/2 ≤ |D(B − 1)|3/2 |D|

1/2 1/2 3/2 so that |B||D| |Si| < |D(B − 1)| , and the bound (2.18) holds. Next we assume that the 2 2 3 second condition in (2.19) does not hold, that is, |B||D(B − 1)| > |Si| |D| . We again use the trivial bound ∗ 2 E (B,Si) ≤ |B||Si| . Comparing this to our desired bound, we have

|S |3/2|D(B − 1)|3/2 |B||S |2 ≤ i ⇐⇒ |B||D|1/2|S |1/2 ≤ |D(B − 1)|3/2 i |D|1/2 i so that the desired bound would follow from the second inequality above. Using our assumption 2 2 3 |B||D(B − 1)| > |Si| |D| , we know that |B|5/4|D(B − 1)|1/2 |B||D|1/2|S |1/2 < i |D|1/4 and by our assumption that |B| ≤ |D|, we have

|B|5/4|D(B − 1)|1/2 ≤ |D(B − 1)|3/2 |D|1/4

1/2 1/2 3/2 so that we have |B||D| |Si| < |D(B − 1)| as needed.

∗ In all cases the bound on E (B,Si) holds, so that we ﬁnd

1/2 1/2 0 2 2 02 ∗ ∗ 4 |R1| |R2| ∆1∆1 ∆2∆2 E (B,S1)E (B,S2)(log |A|) ∆2∆02|S |3/2|S |3/2|D(B − 1)|3 2 2 1 2 (log |A|)4(log |B|)2 |D| E∗ (A)3|D(B − 1)|3 ≤ 4/3 (log |A|)4(log |B|)2. |D|

18 0 Where the ﬁnal inequality follows as ∆2 and ∆2 correspond to representations of elements 3/2 3/2 2 4/3 P 4/33/2 ∗ 3/2 of S1 and S2 in A/A, so that |S1| ∆2 = |S1|∆2 ≤ x rA/A(x) ≤ E4/3(A) , and similarly for S2. Combining the bounds (2.11), (2.17), and the above, we have

3/2 2 13/2 5/2 3/2 1/2 4 4 ∗ 3/2 17/2 2 |Q| ∆ |B| |A| |D| |C| |AB| |C(A+1)||D(B −1)| E4/3(A) (log |A|) (log |B|) which simpliﬁes to

∗ 0 3 13 5 3 8 2 8 ∗ 3 17 4 E4/3(A ) |B| |A| |D| |C| |AB| |C(A + 1)| |D(B − 1)| E4/3(A) (log |A|) (log |B|) .

0 We know by Lemma 1 that E4/3(A ) E4/3(A), so we have that

|B|13|A|5|D|3|C| |AB|8|C(A + 1)|2|D(B − 1)|8(log |A|)17(log |B|)4 as needed.

19 Chapter 3

Aﬃne Group Energy and Applications

This chapter is composed of a combination of the papers [64] and [53], appearing in Discrete & Computational Geometry and International Mathematics Research Notices respectively. The first is joint work with Oliver Roche-Newton, and the second is joint work with Giorgis Petridis, Oliver Roche-Newton, and Misha Rudnev. We prove a nontrivial energy bound for a finite set of affine transformations over a field F and discuss a number of implications. These include new bounds on growth in the affine group, and a quantitative version of a theorem of Elekes concerning rich lines in grids. We also give a positive answer to a question of Yufei Zhao that for a plane point set P for which no line contains a positive proportion of points from P , there may be at most one line l meeting the set of lines defined by P in O(|P |) points.

3.1 Preliminaries

Let F be a field with multiplicative group F∗. We are particularly interested in the zero characteristic case F = R, or F = Fp, where p is an odd prime (there is no difference as to our results between Fp and a general F with characteristic p). Recall from Chapter 1 that the affine group G := Aff(F) consists of non-vertical and 2 non-horizontal lines in F under composition. Such a line has equation y = g1x + g2 with ∗ 2 (g1, g2) ∈ F × F. We may therefore visualise G as the plane F with the y-axis deleted. The composition of two points is given by the equation

(a, b) ◦ (c, d) = (ac, ad + b) which is precisely the composition of the corresponding lines. The identity element in G is the line y = x, which corresponds to the point (1, 0). G has a normal subgroup U = {(1, b): b ∈ F} and subgroups of the form T = T (x) = Stab(x), which we shall call maximal tori, for the action g(x) = g1x + g2 on F. The tori correspond to ﬁnite slope lines through the identity (1, 0) in the coordinate plane, which is seen by considering the equation x = xg1 + g2 deﬁning the stabilizer, from which it follows that (g1, g2) = (g1, x(1 − g1)) (x here is giving the slope of the line). Their left cosets are given by families of parallel lines with the given slope; their right cosets by families of lines incident to the deleted y-axis at the point where the line corresponding to T itself meets it. Cosets of U are vertical lines. For A ⊆ G, we use the notation Ak for the k-fold Cartesian product of A with itself, not to be confused with A−1, the set of inverses of elements of A or the product sets A−1A, AA,

20 etc. We deﬁne two versions of the energy of A as

E(A) := |{(g, h, u, v) ∈ A4 : g−1 ◦ h = u−1 ◦ v}| ,E∗(A) := |{(g, h, u, v) ∈ A4 : g ◦ h = u ◦ v}| . (3.1) These are diﬀerent quantities. From here onwards, we shall omit the symbol ◦ and simply write gh for the product of g and h in Aﬀ(F). A standard application of the Cauchy-Schwarz inequality yields the following bounds connecting E(A) and E∗(A) with A−1A and AA, respectively.

|A|4 |A|4 E(A) ≥ ,E∗(A) ≥ . (3.2) |A−1A| |AA|

In fact, Shkredov [79, Section 4] proved the inequality E∗(A) ≤ E(A). Also, for a scalar set S ⊆ F we use the standard notations E+(S) (and respectively E×(S)) for its additive (and respectively multiplicative) energy. That is,

+ 4 E (S) := |{(s1, s2, s3, s4) ∈ S : s1 + s2 = s3 + s4}|

∗ 4 E (S) := |{(s1, s2, s3, s4) ∈ S : s1s2 = s3s4}|. Note that in principle one cannot expect to have a nontrivial upper bound on E(A), since A can lie in a coset of an Abelian subgroup. The paradigm of the rich theory of growth in groups, developed in the past 15 years, is that this is the only case when A would not grow by multiplication.

In the affine group case, the first nontrivial instance of a non-commutative group, studied among others in [69] (see the references therein) the following incidence theorem was proved, establishing an explicit connection between energy in the affine group and incidence theory.

Theorem 12. Let S,T ⊆ F be ﬁnite sets and A a ﬁnite set of non-vertical, non-horizontal lines in F2. If F = R, then I(S × T,A) |T |1/2|S|2/3E1/6(A)|A|1/3 + |T |1/2|A| . (3.3)

If F = Fp, then

I(S × T,A) |T |1/2|S|5/8E1/8(A)|A|1/2 + |T |1/2|A| · pmax{1, |S|2/p} . (3.4)

Theorem 12 (along with other results) enabled [69, Theorem 7], which is a somewhat stronger version of the following classical theorem of Elekes [23] on rich lines in grids.

Theorem 13 (Elekes). Let α ∈ (0, 1), n be a positive integer, A ⊆ R with |A| = n and suppose there are n non-horizontal and non-vertical lines intersecting A×A in at least αn points. Then either • Ω(αC n) of these lines are parallel, or • Ω(αC n) of these lines are incident to a common point. Here C > 0 is an absolute constant.

As mentioned above, parallel and concurrent lines are in correspondence with coset families in Aff(R) and hence Theorem 13 is a result on affine transformations. Theorem 13 and underlying ideas were further developed by several authors, having inspired Murphy’s work [48], which deals with Aff(F), F being C or finite.

21 Theorem 13 is qualitative in the sense that its proofs so far failed to yield a reasonable quantitative value of C, for the proofs involved a graph-theoretical Balog-Szemer´edi-Gowers type argument, which is rather costly for quantitative estimates. However, it would not be diﬃcult to derive Theorem 13 as a corollary of Theorem 12, provided that there is a nontrivial bound on the energy E(A) therein. Such estimates were obtained in [69] in the special case of A = C × D being a Cartesian product. The bound obtained in [69] was that for some δ > 0,

E(C × D) |C|5/2−δ|D|3, (3.5) provided that the sizes of C and D are not vastly diﬀerent. The quantitative lower bound on δ would be, if attempted, quite small. On the other hand, the above estimate with δ = 0 can be regarded in a sense as a threshold one, and having any δ > 0 makes it strong enough to reach the point at which new information in the arithmetic setting of sum-product theory is obtained, where incidence conﬁgurations involving such Cartesian products frequently arise. Several such applications were explored in [64], and also in [69].

3.2 Main Results

We prove a generalisation of the threshold version of estimate (3.5) for any finite set A ⊂ Aff(F). This enables us to develop a number of geometric combinatorics estimates, which so far have been inaccessible. Our main result is the following: Theorem 14. Let A ⊆ Aff(F) be a finite set, considered as a set of points in F2. Suppose that no line contains more than M points of A, and no vertical line contains more than m points of A. Also, suppose in positive characteristic p, one has m|A| ≤ p2. Then,

∗ 1 5 2 max{E(A),E (A)} m 2 |A| 2 + M|A| .

Note that if A is taken to be a Cartesian product, this recovers (3.5), with δ = 0. We also give a result which improves the bound above for a special case of the form of A ⊆ R. Theorem 15. Let C,D ⊂ R be ﬁnite sets, and let A ⊂ Aﬀ(R) be the points of the form λ µc R c−d , c−d with c ∈ C, d ∈ D, c 6= d, and λ, µ ∈ \{0}. Then we have E(A) |C|5/2|D|5/2 + |C|4 + |D|4.

We remark that one may consider the asymmetric case of E(A, B) for A, B ⊂ Aﬀ(F), deﬁned similar to E(A) in (3.1), only now g, u ∈ A; h, v ∈ B. Give the notations m, M of Theorem 14 subscripts A or B, depending on whether they pertain to A or B. The estimate of Theorem 14 then changes to

1 3 2 2 E(A, B) mA|A||B| + min{MA,MB}|A||B| ,

2 provided that mA|B| ≤ p in positive characteristic. Moreover, if F = Fp one can dispense with the p-constraint, by using the asymptotic version of Theorem 19, estimate (3.14). This would 2 mA|A||B| add to the right-hand side of the latter estimate a (crudely estimated) term p . What is the correct bound for E(A)? The example A = C × D, where C is a geometric progression and D an arithmetic progression, with |D| > |C| shows, in view of the forthcoming

22 equations (3.12), that in this case E(A) M|A|2. Note that owing to energy quadruples (g, h, u, v) lying in the same (left) coset of some torus, the upper bound on E(A) should exceed the number of collinear point triples in A. We carefully conjecture that M|A|2 may be the correct asymptotics, possibly with additional subpolynomial factors in |A|, although we do not have evidence for their presence. Something similar has been conjectured for the concept of bisector energy in [44]. The proof of Theorem 14 is geometric, relying on incidence theory. After an application of Cauchy-Schwarz in the form of (3.2), it enables an estimate on growth in Aﬀ(Fp) which is much stronger than its algebra-based predecessors in [38, Proposition 4.8], [48, Theorem 27] and even the geometric [69, Theorem 5] which took advantage of the (sharp) estimate on the F2 number of directions determined by a point set in p by Sz˝onyi [85].

Corollary 2. Let A ⊂ Aﬀ(Fp). Suppose that no line contains more than M points of A, and no vertical line contains more than m points of A. Suppose m|A| ≤ p2. Then

min{|AA|, |A−1A|} m−1/2|A|3/2 + M −1|A|2 .

In particular, if |AA| = K|A|, then A has |A|/K2 elements in a coset of U or |A|/K elements in a coset of a torus.

Theorem 14 enables us to establish the following quantitative version of Theorem 13 over general ﬁelds.

Theorem 16. Let n, k be positive integers, A ⊂ F with |A| = n and suppose there are k non- − 1 horizontal and non-vertical lines intersecting A×A in at least αn points for some n 2 α ≤ 1. In positive characteristic p also assume

p max{k, α−2n} .

Then for C = 12 if F = R and C = 16 if F = Fp, we have either • Ω(αC n−2k3) of these lines are parallel, and moreover E+(A) α2+C k3; or • Ω(αC/2n−1k2) of these lines are incident to a common point, and moreover there is s ∈ F × 2+ C 2 such that E (A − s) α 2 nk .

An Application to Line Geometry

Theorem 16 gives an aﬃrmative answer (with a quantitative estimate) to a conjecture stated in [64, Conjecture 8] – a question that had earlier been raised by Yufei Zhao. We give a statement over the reals, although it applies to a general F, given that P deﬁnes Ω(|P |2) lines. Given a planar point set P and a line l not incident to P , we call the shadow of P on l the set of points where l meets the lines from L(P ) – the set of lines, generated by pairs of points in P . Abusing notation slightly, we write L(P ) ∩ l for the shadow of P on l.

2 Corollary 3. Let P ⊂ P(R ) be a ﬁnite, suﬃciently large set of points, and let l1 and l2 be two distinct lines such that |P |1−δ1 points of P lie on any line passing through the intersection 1−δ2 of l1 and l2, and |P | points of P lie on any other line, for some δ1, δ2 > 0. Then

1+ δ1 1+ δ2 max{|L(P ) ∩ l1|, |L(P ) ∩ l2|} min{|P | 14 , |P | 7 }.

23 Therefore, a suﬃciently non-collinear point set P can have a small shadow on at most one line. This question also has connections to the sum-product problem. Indeed, if we take 2 the point set to be P = {(a, a ): a ∈ A ⊂ R}, l1 to be the line at inﬁnity and l2 to be the y-axis, Corollary 3 implies that

|A + A| + |AA| |A|1+c ,

1 with c = 14 . This implication is true for sufficiently small sets in positive characteristic as well, for the role of R in the proof of Corollary 3 is merely to guarantee that |L(P )| |P |2, via Beck’s theorem. But in fact, if P is a parabola as considered above, it’s easy to show that (as a set of affine transformations) E(P ) |P |2, and following the argument in the proof of Corollary 3, 1 1 one finds exponents of c = 7 in characteristic zero and c = 9 in positive characteristic. We use Corollary 3 to point out another curious fact, concerning the shadow of the Cartesian product A × A on a single line.

Corollary 4. Let A ⊂ R be a ﬁnite set, and P = A × A. Let l be any aﬃne line not of the form y = x or y = −x + k for some k. Then we have

2+ 1 |L(A × A) ∩ l| |A| 14 .

3.2.1 Applications to the Sum-Product Phenomenon

Our main applications of the results above concerns the sum-product phenomenon, see Chapter 1 for a basic introduction. It was proven by Elekes and Ruzsa [27] that

|A|6 |A + A|4|AA| , log |A| and in particular it follows that

|A|2 |A + A| ≤ K|A| ⇒ |AA| . (3.6) K4 log |A|

A version of (3.6) with a better dependency on K follows from the beautiful work of Solymosi [82]. This problem is often referred to as the ’few sums - many products’ problem. (3.6) implies that the Erd˝os- Szemer´ediconjecture holds in the extreme case |A + A| |A|. We consider an asymmetric version of this question, where the ultimate goal would be to prove a result of the form

|A||B| |A + A| ≤ K|A| ⇒ |AB| , (3.7) KC log |A| for any B ⊂ R ﬁnite and some constant C. Such a result would have some striking implications, and appears to be out of reach at present. It follows from [24] (and also from [82]) that

|A||B|1/2 |A + A| ≤ K|A| ⇒ |AB| . (3.8) K Equation (3.8) therefore represents the ‘threshold bound’ for this problem. We present the following small improvement.

24 Theorem 17. For all κ > 0 there exists k = k(κ) > 0 such that for all A, B ⊂ R with |A|κ ≤ |B| and writing |A + A| = K|A|, we have

|A||B|1/2+k |AB| . K4/3

The second result we give concerns expanders. Several expander results, including the bound |A|2 |A(A + A + A + A)| , (3.9) log |A| were given by Murphy, Roche-Newton and Shkredov in [50]. Note that this quadratic lower bound is optimal up to logarithmic factors, as can be seen by taking A to be an arithmetic progression. The same paper also included the bound ad − bc 2 : a, b, c, d ∈ A |A| . (3.10) a − c

ad−bc As mentioned in Chapter 1, the quantity a−c has some geometric meaning: if we draw a straight line through (a, b) and (c, d) with a 6= c, this line will intersect the y-axis at the point ad−bc 0, a−c . So the set in (3.10) is the set of all y-intercepts of lines determined by A × A. In this Chapter, we give the following improvement to (3.10).

Theorem 18. Let A ⊂ R be a ﬁnite set. Then ac − db 2+1/14 : a, b, c, d ∈ A |A| . c − d

This is an example of a four-variable super-quadratic expander. Several six-variable superquadratic expanders were proven to exist by Balog, Roche-Newton and Zhelezov [5]. Typically things get more diﬃcult with less variables, and examples of super-quadratic four variable are rare in the literature. We are aware of only two such results, due to Rudnev [66] and Shkredov [78].

3.3 Proofs

3.3.1 Proof of Theorems 14 and 15

We adopt the convention that for any x ∈ Aﬀ(F), x = (x1, x2). We shall ﬁrst prove Theorem 14.

Proof of Theorem 14. We prove the estimate for the quantity E(A), the proof for E∗(A) follows from the inequality E∗(A) ≤ E(A) from [79]. Alternatively the forthcoming proof has an isomorphic version with E∗(A) in place of E(A). Set Q(A) := {(g, h, u, v) ∈ A4 : g−1 ◦ h = u−1 ◦ v}, (3.11) so that E(A) = |Q(A)|. The ﬁrst lemma gives a convenient decomposition of this set.

25 Lemma 2. Let A ⊂ G be ﬁnite. Then we have X E(A) = |{(g, h, u, v) ∈ Q(A): g1v1 = C = h1u1}|. C

Proof. Suppose that g−1 ◦ h = u−1 ◦ v. Then

−1 −1 −1 −1 (g1 , g2g1 ) ◦ (h1, h2) = (u1 , u2u1 ) ◦ (v1, v2).

Calculating the ﬁrst coordinates then gives

−1 −1 h1g1 = v1u1 which rearranges to g1v1 = h1u1.

Let QC denote the set of solutions corresponding to the value C in the decomposition above, that is, QC := |{(g, h, u, v) ∈ Q(A): g1v1 = C = h1u1}|. Also deﬁne the set PC := {(g, v) ∈ A × A : g1v1 = C}.

We bound QC by the following lemma. Lemma 3. For any C ∈ F∗, under the constraint that m|A| ≤ p2 in positive characteristic, one has 3/2 QC |PC | + M|PC |.

Proof of Lemma 3. By considering the line equation g−1 ◦ h = u−1 ◦ v and expanding out, we have the equations h2 − g2 v2 − u2 g1v1 = h1u1 = C, = . (3.12) g1 u1 The second equation can be written as

u2g1 − u1g2 − g1v2 + u1h2 = 0 . (3.13)

3 3∗ 3 We therefore map PC × PC to P × P (where P is the three-dimensional projective space over F and P3∗ is its dual) as

(g, v) → (g1 : g2 : g1v2 : 1), (u, h) → (u2 : −u1 : −1 : u1h2) .

0 0 0 0 The map is injective. Indeed, suppose we have (g1, g2, g1v2) = (g1, g2, g1v2). Then we must have 0 0 0 0 0 (g1, g2) = (g1, g2), and since g1v1 = g1v1 = C, we must also have v1 = v1. The last coordinate 0 0 0 then gives g1v2 = g1v2 =⇒ v2 = v2. Thus equation (3.13) can be viewed as a point plane incidence between the set of points

PC = {(g1, g2, g1v2):(g, v) ∈ PC } and the set of planes ΠC = {πu,h :(u, h) ∈ PC }, where πu,h is the plane given by u2x−u1y−z+u1h2 = 0. Note that we have |PC | = |ΠC | = |PC |. This calls for using a point-plane incidence bound from [67]:

26 Theorem 19 (Rudnev). Let F be a field, and let P and Π be finite sets of points and planes respectively in P3. Suppose that |P | ≤ |Π|, and that |P | p2 if the characteristic p 6= 2 of F is positive. Let k be the maximum number of collinear points in P . Then the number of incidences satisfies I(P, Π) |Π||P |1/2 + k|Π|.

If F = Fp, one can dispense with the p-constraint in Theorem 19, replacing its estimate by the following asymptotic version (see, e.g. [70, Lemma 6]):

|Π||P | I(P, Π) − |Π||P |1/2 + k|Π| . (3.14) p

We claim that we can set k = M in our application of Theorem 19. Indeed, suppose that a line contains more than M points from PC , each of the form (g1, g2, g1v2). If this line is not of the form {(x0, y0, z): z ∈ F} then project using the map π(x, y, z) = (x, y, 0) to get a line in the plane which contains more than M points of the form (g1, g2). This contradicts the assumption that A contains no more than M points on a line.

If the line is of the form {(x0, y0, z): z ∈ F} and it contains more than M points from PC , it follows that there are more than M points (v1, v2) in A, all satisfying v1 = C/x0.

To conclude the proof we apply Theorem 19 to the point-plane arrangement PC and ΠC . We note the trivial (yet easily seen to be achievable) supremum bound

|PC | ≤ m|A| , (3.15) which determines the constraint in positive characteristic, and then have

3/2 |QC | ≤ I(PC , ΠC ) |PC | + M|PC | , as required.

The claim of Theorem 14 now follows immediately. Apply Lemma 2 and bound each summand by Lemma 3 to get

X 3/2 E(A) ≤ |PC | + M|PC | . C

Applying the L∞ bound (3.15) and the L1 identity

X 2 |PC | = |A| C completes the proof.

Lemma 3 cannot be improved without additional restrictions. Indeed, consider the case 2 A = [n]×[n]. If we take C = q for some prime q ≤ n, we have |PC | = 2n . We also have M = n 1/2 (hence M is a constant multiple of |PC | - the regime in which the two terms in Lemma 3 are 3 of the same order of magnitude). Lemma 3 tells us that QC n . On the other hand, every solution to the additive energy equation

b1 + b2 = b3 + b4, bi ∈ {1, . . . , n}

27 3 gives a contribution to QC , and so QC n . In fact, if we instead take C to be any element of the product set of [n], then since C 2+ has n representations as a product we have |PC | n , and so the bound is near-optimal for all C for this choice of A. Moreover, estimate (3.15) alone is also sharp if one simply takes A = C×D, where C is a geometric progression. Thus the coarseness in the estimate of Theorem 14 is rooted, not surprisingly, in the sum-product phenomenon. We now prove Theorem 15. We will make use of the following simple corollary of the Szemerédi-Trotter theorem concerning the number of k rich lines defined by a finite point set. 2 Corollary 5. Let P ⊂ R be a finite set of points. Let k ≥ 2 and let Lk(P ) denote the set of all lines containing at least k points from P . Then we have |P |2 |P | |L (P )| + . k k3 k

Proof of Theorem 15. Recall that the aim is to bound the energy of the set of points λ µc A := , : c ∈ C, d ∈ D, c 6= d . c − d c − d c−d µ Inverses of points in A have the form λ , − λ c . The quantity E(A) is therefore the number of solutions to the equations c − d c − d 1 1 = 3 3 (3.16) c2 − d2 c4 − d4 c d − c d c d − c d 1 2 2 1 = 3 4 4 3 . (3.17) d2 − c2 d4 − c4

Note that equation (3.16) asserts that the line connecting the points (c2, c1) and (d2, d1) has the same slope as the line connecting the points (c4, c3) and (d4, d3), i.e. they are parallel. Equation (3.17) then asserts that these two lines in fact have the same y-axis intercept. Indeed, upon calculating the equations of the lines connecting the pairs of points above, the y intercepts are precisely c1d2−c2d1 and c3d4−c4d3 . Therefore the lines are the same, and we have a collinear d2−c2 d4−c4 quadruple (c2, c1), (d2, d1), (c4, c3), and (d4, d3). Hence the energy E(A) is no larger than the 2 2 number of collinear quadruples (p1, p2, p3, p4) ∈ (C ×C) ×(D ×D) . Our goal is now to bound the number of such quadruples. We begin with the trivial observation that there are at most |C|2|D|2 such quadruples with p1 = p2 and p3 = p4. Therefore, X E(A) ≤ |C|2|D|2 + |l ∩ (C × C)|2|l ∩ (D × D)|2. lines l:max{|l∩(C×C)|,|l∩(D×D)|}≥2 We ﬁrstly separate this sum as X X |l ∩ (C × C)|2|l ∩ (D × D)|2 + |l ∩ (C × C)|2|l ∩ (D × D)|2. (3.18) lines l: lines l: |l∩(C×C)|≥2, |l∩(C×C)|=1 |l∩(D×D)|≥2 XOR |l∩(D×D)|=1 To deal with the second term, note that X X X |l ∩ (C × C)|2|l ∩ (D × D)|2 = |l ∩ (D × D)|2 + |l ∩ (C × C)|2 lines l: l:|l∩(C×C)|=1 l:|l∩(D×D)|=1 |l∩(C×C)|=1 |l∩(D×D)|≥2 |l∩(C×C)|≥2 XOR |l∩(D×D)|=1 |D|4 + |C|4.

28 We now bound the ﬁrst summand in (3.18). By the Cauchy-Schwarz inequality,  1/2 X 2 2  X 4 |l ∩ (C × C)| |l ∩ (D × D)| ≤  |l ∩ (C × C)|  (3.19) lines l: lines l: |l∩(C×C)|≥2 |l∩(C×C)|≥2 |l∩(D×D)|≥2  1/2  X 4 ·  |l ∩ (D × D)|  (3.20) lines l: |l∩(D×D)|≥2 so that we may instead separately bound the number of quadruples from C × C and D × D. To do this, one can dyadically decompose and apply Corollary 5 as follows:

log |C| X X X |l ∩ (C × C)|4 = |l ∩ (C × C)|4 lines l: j=1 l:2j ≤|l∩(C×C)|<2j+1 |l∩(C×C)|≥2 log |C| X 2j|C|4 + |C|223j |C|5. j=1 The same bound holds for the corresponding term for D in (3.19). Putting everything together, it follows that E(L) |C|5/2|D|5/2 + |C|4 + |D|4, as required.

3.3.2 Proof of Theorem 16

Proof. We apply Theorems 12 and 14. The constraints on α in the statement of Theorem 16 have been chosen to ensure dominance of the ﬁrst terms in the incidence bounds present in 12. Let L be the set of k lines described in the statement of Theorem 16. For F = R an application of Theorems 12 and 14 combined with the fact that αn ≥ C0n1/2 gives αkn ≤ I(A × A, L) n7/6 m1/12k3/4 + M 1/6k2/3 , so either m α12n−2k3 or M α6n−1k2. It follows that either Ω(α12n−2k3) lines, as elements of G, lie in the same coset of U, so they are parallel, or Ω(α6n−1k2) lines lie in the same coset of some torus, so they are concurrent.

Similarly, for F = Fp, Theorems 12 and 14 combined (the condition k ≤ p ensures that mk ≤ p2) with the fact that αn ≥ C0n1/2 and p ≥ α−2n give αkn n9/8 m1/16k13/16 + M 1/8k3/4 , hence either Ω(α16n−2k3) lines are parallel or Ω(α8n−1k2) lines are concurrent. To address the second claim of the first bullet point, parameterise the set of parallel lines by (γ, β), where γ 6= 0 is the fixed slope, and denote by B ⊆ Fp the set of all such β. By the assumption that we are in the case of the first bullet point, we have |B| αC n−2k3 ,

29 where C = 12 or C = 16. Hence, applying Cauchy-Schwarz twice, one has (with rF (A)(β) standing for the number of realisations of β as a value of a function F , with several variables in A, and A(x) being the indicator function of A):

!1/2 X X X p X 2 αn|B| ≤ A(γx + β) = rA−γA(β) ≤ |B| rA−γA(β) β∈B x∈A β∈B β∈B (3.21) ≤ p|B|pE+(A, γA) ≤ p|B|pE+(A) , proving the second claim of the ﬁrst bullet point.

C/2 −1 2 Similarly in the case of Ω(α n k ) lines being concurrent at some point (x0, y0), we parameterise them by their slopes, once again denoted as β = y−y0 in a set B, with |B| x−x0 αC/2n−1k2. As in (3.21) above, we have

X 1 p × × 4 αn|B| ≤ r A−y0 (β) ≤ |B|(E (A − x0)E (A − y0)) , A−x0 β∈B proving the second claim of the second bullet point.

3.3.3 Proof of Corollaries 3, 4, and Theorem 18

In this section we ﬁrst apply Theorem 14 to prove Corollary 3, whence we deduce Corollary 4 and Theorem 18.

Proof of Corollary 3. Let l1, l2 be distinct arbitrary lines, and let P be a set of points in the 1−δ1 plane with no more than |P | points of P on a line through the intersection of l1 and l2, and no more than |P |δ2 points on any other line. By assumption, no more than |P |1−δ1 points lie on l1 or l2, so we may remove such points and be left with a positive proportion of P . We apply a projective transformation π to send l1 to the y-axis ly, and l2 to the line at inﬁnity l∞. Deﬁning P 0 := π(P ), we have

0 0 |L(P ) ∩ l1| = |L(P ) ∩ ly|, |L(P ) ∩ l2| = |L(P ) ∩ l∞| which follows since projective transformations preserve incidence structure. Let T denote the 0 0 set L(P ) ∩ ly and S denote the set L(P ) ∩ l∞. Note that lines through the intersection of l1 and l2 have been mapped to vertical lines. We claim that we have I(P 0,L(P 0)) ≤ I(S × T,P 0) where on the right side P 0 is being considered as a subset of the affine plane Aff(R) corresponding 0 to a set of lines. The claim follows by taking a point (p1, p2) ∈ P lying on some line y = sx + t coming from L(P 0). By definition we must have (s, t) ∈ S × T , and therefore this incidence may be viewed as the point (s, t) ∈ S × T lying on the line y = −p1x + p2, proving the claim. We bound these incidences using Theorem 12, applied to the point set S × T and the 0 0 line set P (strictly we are applying it to the line set given by (−p1, p2) such that (p1, p2) ∈ P , but this changes nothing with respect to the collinearity conditions). The lines in P 0 are non- vertical and non-horizontal since we removed any points lying on l1 or l2 in P , which implies

30 that P 0 has only aﬃne points lying oﬀ the y axis. Furthermore, P 0 has no more than |P |1−δ1 points on any vertical line, and no more than |P |1−δ2 points on any other line. We may bound the energy E(P 0) using Theorem 14. Plugging in m |P 0|1−δ1 , M |P 0|1−δ2 , we have 0 0 3− δ1 0 3−δ E(P ) |P | 2 + |P | 2 , implying that

I(S × T,P 0) |T |1/2|S|2/3E(P 0)1/6|P 0|1/3 + |T |1/2|P 0| 1/6 1/2 2/3 0 3− δ1 0 3−δ 0 1/3 1/2 0 |T | |S| |P | 2 + |P | 2 |P | + |T | |P |

1/2 2/3 0 5 − δ1 1/2 2/3 0 5 − δ2 1/2 0 |T | |S| |P | 6 12 + |T | |S| |P | 6 6 + |T | |P |.

Since each line in L(P 0) intersects P 0 in at least two places and by Beck’s theorem [7] we have |L(P 0)| |P 0|2, we may bound below by

I(S × T,P 0) ≥ I(P 0,L(P 0)) ≥ |L(P 0)| |P 0|2.

Putting this all together, we have

0 2 1/2 2/3 0 5 − δ1 1/2 2/3 0 5 − δ2 1/2 0 |P | |T | |S| |P | 6 12 + |T | |S| |P | 6 6 + |T | |P |.

If the third term dominates we have a stronger result, so we may assume one of the ﬁrst terms dominates, giving

0 1+ δ1 0 1+ δ2 max{|L(P ) ∩ l1|, |L(P ) ∩ l2|} = max{|S|, |T |} min{|P | 14 , |P | 7 }.

This gives the result since |P | = |P 0|.

Proof of Corollary 4. Let A ⊆ R be a finite set, and l an affine line not of the form y = −x + k 2+ 1 for some k, or the line y = x. We shall prove that |L(A × A) ∩ l| |A| 14 . Let γ denote the transformation given by reflection in the line y = x. The Cartesian product A × A is sent to itself under this reflection. Furthermore, by the restriction placed on the line l, we must have that l0 := γ(l) 6= l. Since A × A is symmetric under γ, it follows that L(A × A) is also symmetric under γ, and moreover that the number of intersections of L(A × A) with l is precisely the number of intersections of L(A × A) with l0, i.e. |L(A × A) ∩ l| = |L(A × A) ∩ l0|. An application of Corollary 3 with δ1 = δ2 = 1/2 then gives

0 2+ 1 |L(A × A) ∩ l| = max{|L(A × A) ∩ l|, |L(A × A) ∩ l |} |A| 14 .

Note that one may consider a more general version of Corollary 4, applying to a point set in R2 with a symmetry. We can now immediately prove Theorem 18, by applying Corollary 4 with the line l being the y-axis. Together with the observation made above that L(A × A) intersects the y-axis at ad−bc the points a−c proves the theorem.

31 3.4 Sum-Product Applications

3.4.1 Asymmetric ‘Few Sums Many Products’ Problem

The purpose of this section is to prove Theorem 17. We will need a bound of the form (3.5) for the energy of lines of the form (a, ab). In fact, such a result is already provided in [69, Lemma 21], which we state below.

Theorem 20. For all κ > 0 there exists δ = δ(κ) > 0 such that, for all C,D ⊂ R∗ with |D|κ ≤ |C| ≤ |D|2, the set of lines

L = {(c, cd):(c, d) ∈ C × D} ⊂ Aﬀ(R) satisﬁes the bound 5 −δ 3 E(L) |C| 2 |D| .

The proof of Theorem 20 is significantly more difficult than that of (3.5), utilising bounds on growth in the affine group proved elsewhere in [69], as well as an additive combinatorial tool due to Shkredov [77] which gives structural information for a set when its second moment and third moment energy are in a particular ‘critical case’. Theorem 20 can then be combined with Theorem 12 to give the following incidence theorem.

Theorem 21. For all κ > 0 there exists δ = δ(κ) > 0 such that, for all C,D ⊂ R∗ with |D|κ ≤ |C| ≤ |D|2, the set of lines

L = {(c, cd):(c, d) ∈ C × D} ⊂ Aﬀ(R) satisﬁes the bound

I(A × B, L) |A|2/3|B|1/2|C|3/4−δ|D|5/6 + |B|1/2|C||D| for any A, B ⊂ R.

We will also need to know that sets with small sum set have superquadratic sized triple product sets. The precise statement we use is [62, Theorem 3.2], stated below.

Theorem 22. There is an absolute constant C > 0 such that, for any A ⊂ R and |A + A| = K|A| we have 2+ 1 |A| 392 |AAA| 125 . K 56 (log |A|)C

We are now ready to begin the proof of Theorem 17, which we restate below for convenience.

Theorem 17. For all κ > 0 there exists k = k(κ) > 0 such that for all A, B ⊂ R ﬁnite with |A|κ ≤ |B| and writing |A + A| = K|A|, we have

|A||B|1/2+k |AB| . K4/3

32 Proof. Fix κ > 0 and let δ = δ(κ) be the value given by the statement of Theorem 21. Firstly, κ 2−c0 8δ 1 let us assume that |A| ≤ |B| ≤ |A| , where c0 = min{ 1+4δ , 392 }. For this case, we can use a simple Elekes-type argument in combination with Theorem 21. Let L be the set of all lines of the form y = c(x − d) with c ∈ B and d ∈ A. Using the notation of Aﬀ(R), this is the set of all lines of the form (c, −cd). Deﬁne P = (A + A) × (AB). Observe that, for each a ∈ A and for any (c, d) ∈ A × B we have an incidence between the point (a + d, cd) and the line (c, −cd). Applying this observation and Theorem 21 yields |A|2|B| ≤ I(P, L) |A + A|2/3|AB|1/2|B|3/4−δ|A|5/6 + |AB|1/2|B||A| |A + A|2/3|AB|1/2|B|3/4−δ|A|5/6,

2 where the latter inequality uses the assumption that |B| ≤ |A| 1+4δ . Writing |A + A| = K|A| and rearranging gives 1 +2δ |A||B| 2 |AB| 4 . (3.22) K 3 Note also that Theorem 17 is true for trivial reasons when |B| ≥ |A|3. This is simply because for any sets A and B with this property 2 |AB| ≥ |B| ≥ |A||B| 3 .

Now we turn to the case when |A|2−c0 ≤ |B| ≤ |A|3. Applying Theorem 22 and the Pl¨unnecke-Ruzsa Theorem, gives 3 2+ 1 |AB| |A| 392 2 ≥ |AAA| 125 . |B| K 56 (log |A|)C Rearranging and using the bound |B| ≥ |A|2−c0 , it follows that 2 2 + 1 1 1+ 1 − c0 1 1+ 1 1 1+ 1 |B| 3 |A| 3 1176 |B| 2 |A| 1176 6 |B| 2 |A| 2352 |B| 2 |A| 2500 |AB| 125 125 ≥ 125 4 . K 168 (log |A|)C K 168 (log |A|)C K 168 (log |A|)C K 3 1 In the penultimate inequality above we have used the fact that c0 ≤ 392 , while the last step is trivially true (for A suﬃciently large) and is carried out only to simplify the expression. Finally, since |A| ≥ |B|1/3, we conclude that 1 + 1 |B| 2 7500 |A| |AB| 4 . (3.23) K 3 Examining inequalities (3.22) and (3.23), we see that the proof is complete by taking k = 1 min{2δ, 7500 }.

We remark that the arguments from proof of Theorem 20 can be applied to bound the energy of lines of the form (cd, c). One can then repeat some arguments from the proof of Theorem 17 above but using this improved energy bound in order to get a threshold breaking sum-product type result of the following form: 1 +c |AA| ≤ K|A| ⇒ |(A + 1)B| K |A||B| 2 . However, this result does not apply in the endpoint case when the size of B is very close to |A|2, since we do not have a suitable analogue of Theorem 22 for this problem. It may be an interesting research problem to prove such a result, which take the form of the following superquadratic growth statement: |AA| ≤ K|A| ⇒ |(A + 1)(A + 1)(A + 1)| |A|2+c.

33 3.4.2 The size of AA + A

Our next application concerns the size of the set AA + A. The ‘threshold’ bound |AA + A| |A|3/2 follows from a simple application of the Szemer´edi-Trotter theorem. In [61], a fairly involved argument based upon the geometric setup of Solymosi [82] was used to give the small improvement |AA + A| |A|3/2+c, with c = 2−222. This argument required the restriction thatAconsists only of positive reals. In [69], the authors observed that they could remove this restriction and implicitly give a better value of c, although they did not calculate c explicitly. They also obtained an improvement to the corresponding threshold energy bound. The following result gives the ﬁrst ‘reasonable’ explicit bound for this set.

Theorem 23. Let A ⊂ R be ﬁnite. Then

|AA + A| & |A|3/2+1/194.

The proof uses a combination of Theorem 12 and the theory of the quantity d∗(A), in addition to additive and multiplicative energies. Recall that the additive energy of a set A is deﬁned as follows: + 4 E (A) := (a, b, c, d) ∈ A : a + b = c + d . For an integer k, the k’th multiplicative energy of a set A is deﬁned as ∗ 2k a1 a3 a2k−1 Ek(A) := (a1, a2, ..., a2k) ∈ A : = = ... = . a2 a4 a2k

In the case k = 2 this is simply called the multiplicative energy and denoted by E∗(A). We deﬁne the following quantity.

|Q|2|R|2 d∗(A) := min min . t>0 Q6=∅,R⊂R\{0} |A|t3

The following result is stated in [51], and proven in a diﬀerent form in [77].

Theorem 24. Let A ⊂ R be a ﬁnite set. Then we have

+ 7/13 32/13 E (A) . d∗(A) |A| .

A second result concerning d∗(A) is the following decomposition type theorem, which can be viewed as a refinement of the Balog-Szemerédi-Gowers Theorem tailored towards a specific sum-product application. See [51, Lemma 6.4].

R ∗ |A|3 0 Lemma 4. Let A ⊂ be ﬁnite, and let E (A) K . Then there exists A ⊆ A and a number |A| ≥ ∆ |A|/K such that

|A|2 K|A0|2 |A0| & , d (A0) . . K∆ ∗ |A|∆

We are now ready to begin the proof.

34 Proof. It can be assumed without loss of generality that 0 ∈/ A. Write la,b for the line with equation y = ax + b and deﬁne L = {la,b : a, b ∈ A} We begin by bounding the energy of L. The following reﬁnement of (3.5) was given in [69]: ∗ 1/2 1/2 E(L) ≤ E4 (A) Q . Here Q denotes the number of solutions to the equation

a1 − a2 a5 − a6 = , ai ∈ A. a3 − a4 a7 − a8 The bound Q |A|6 log |A| was established in [59] (a simpler proof was later given in [50] - see also [9] for another presentation of this proof).

∗ 2 ∗ Using this result and the trivial bound E4 (A) ≤ |A| E (A), E(L) may then be bounded by E(L) . |A|4E∗(A)1/2. We now use Theorem 12. Define P = A × (AA + A). Since for every c ∈ A we have the 3 incidence (c, ac + b) ∈ la,b, it follows that there are at least |A| incidences. We bound the other side via Theorem 12 as |A|3 ≤ I(P, L) |AA + A|1/2|A|2E∗(A)1/12 + |AA + A|1/2|A|2. Note that we may assume the leading term dominates, as otherwise we do better. We then have |A|2 |AA + A| . E∗(A)1/6 Now define K via E∗(A) = K−1|A|3, so 1 ≤ K ≤ |A|. The above inequality gives |AA + A| K1/6|A|3/2. (3.24) 0 0 & |A|2 0 . K|A0|2 We shall now apply Lemma 4 to A, to find a subset A with |A | K∆ and with d∗(A ) |A|∆ for some |A| ≥ ∆ |A|/K. Applying Theorem 24 gives E+(A0)13/7 K|A0|2 . d (A0) . |A0|32/7 ∗ |A|∆ and so K7/13|A0|46/13 E+(A0) . . ∆7/13|A|7/13 0 + 0 + 0 0 |A0|4 It follows from Cauchy-Schwarz that for any a ∈ A we have E (A ) ≥ E (A , aA ) ≥ |A0+aA0| . Noting that |A0 + aA0| ≤ |AA + A| then gives |A0|6/13|A|7/13∆7/13 |A|19/13∆1/13 |A|20/13 |AA + A| & & . K7/13 K K14/13 Now note that estimate (3.24) improves as K increases, and the estimate above improves as K decreases. We therefore find the value of K where the two estimates give the same value, as elsewhere we always have a better result. We have |A|20/13 = K1/6|A|3/2 =⇒ K = |A|3/97. K14/13 Plugging this back into the estimates gives |AA + A| & |A|3/2+1/194. as required.

35 3.4.3 Another Three-Variable Expander

A further application of the line energy method gives the following expander result.

Theorem 25. Let A ⊆ R be a ﬁnite set. Then we have

5/3 |{(a1 − a2)a3 + a1 : a1, a2, a3 ∈ A}| |A| .

Note that this is better than the usual ‘threshold’ estimate for three-variables that one obtains from a simple application of the Szemer´edi-Trotter Theorem, where an exponent 3/2 typically appears.

Proof. Theorem 25 is a consequence of an energy bound on the set of lines

L := {(c − d, c):(c, d) ∈ C × D, c 6= d} for arbitrary ﬁnite sets C,D ⊂ R∗. In fact, these lines are precisely the inverses of the lines given in Theorem 15. It is now enough to see that for a set of (non-vertical, non-horizontal) lines L, we have E(L) = E(L−1). Indeed, this is seen via

−1 −1 −1 −1 l1 l2 = l3 l4 ⇐⇒ l2l4 = l1l3 .

Using Theorem 15, we have the bound

E(L) |C|5/2|D|5/2 + |C|4 + |D|4. (3.25)

The lines lc,d ∈ L have the form y = (c − d)x + c. Let A ⊂ R be a ﬁnite set, and deﬁne

S := {(c − d)a + c :(a, c, d) ∈ A × C × D}.

Deﬁne the point set P as A × S. Since for each line lc,d in L and each a ∈ A we have the incidence (a, (c − d)a + c) ∈ lc,d, it follows that I(P, L) ≥ |A||C||D|. Using (3.25) and Theorem 12 then gives |A||C||D| ≤ I(P, L) |S|1/2 |A|2/3|C|3/4|D|3/4 + |A|2/3|C||D|1/3 + |A|2/3|D||C|1/3 + |C||D| .

Taking A = B = C ensures that the leading term dominates, completing the proof.

36 Chapter 4

Pencils of Lines and 4-rich Points

4.1 Introduction

As stated in Chapter 1, we consider a question raised by Rudnev: given four pencils of n concurrent lines in R2 with the four centres of the pencils non-collinear, what is the maximum possible size of the set of points where four lines meet? Our main result states that the number of such points is O(n11/6), improving a result of Chang and Solymosi [16]. The results in this chapter first appeared in [63], excluding Section 4.6 which are unpublished expander results. Recall that an n-pencil with centre p ∈ P2(R) is defined to be a set of n concurrent lines passing through p. Given m n-pencils, a point is said to be m-rich if one line from each of the pencils passes through it. The question we study in this chapter is the following: what is the maximum possible size of the set of m-rich points determined by m n-pencils? The first interesting case is when m = 4. For m = 2, 3 there are natural constructions giving Ω(n2) m-rich points, which is certainly maximal.1 Furthermore, when m = 4 and the centres of the four pencils are collinear, it is still possible2 to give a construction generating Ω(n2) 4-rich points. With these degenerate cases dismissed, we arrive at the following two questions of Rudnev.

Problem 1. Given four n-pencils whose centres do not lie on a single line, what is the maximum possible size of the set of 4-rich points they determine?

Problem 2. Given four n-pencils whose centres are in general position (i.e. no three of the centres are collinear), what is the maximum possible size of the set of 4-rich points they determine?

It is possible that the answers to these two questions are the same.

1For m = 2, any two n-pencils with distinct directions determine exactly n2 crossing points. For m = 3, one can take two of the centres of the pencils on the line at infinity so that their crossing points give a grid A × A where A is a geometric progression. Choosing the origin as the centre for the third pencil, Ω(n2) of the points of A × A can be covered by n lines through the origin by using the ratio set as the set of slopes. 2One way to see this is by taking the four centre points on the line at infinity. The first two pencils again intersect in a grid A × A, and this time we make A = {1, 2, . . . , n}. The second two pencils give a family of lines with slopes 1 and −1 respectively, and both directions give rise to a family of lines of size 2n − 1 which cover A × A. Thus we have four pencils of size O(n) (with their centres collinear) and n2 4-rich points.

37 Some progress on the ﬁrst problem was given in a paper of Alon, Ruzsa and Solymosi [2]. They gave a construction of four n-pencils with non-collinear centres which determine Ω(n3/2) 4-rich points. From the other side, a result of Chang and Solymosi [16] implies that for any four n-pencils with non-collinear centres, the number of 4-rich points is O(n2−δ). Their proof gives the value δ = 1/24. The main results of this chapter are the following two theorems, which give improved upper and lower bounds respectively for the maximum possible number of 4-rich points. Theorem 5. Let P be the set of 4-rich points deﬁned by a set of four non-collinear n-pencils. Then we have |P | = O(n11/6). Theorem 26. There exist four n-pencils with non-collinear centres which determine Ω(n3/2 logc n) 4-rich points, for some absolute constant c > 0.

The construction given in [2] of four pencils determining Ω(n3/2) had three of the centres on a line, and thus it did not immediately give any progress towards Problem 2. We give a similar construction with no three of the centres on a line. Theorem 27. There exist four n-pencils, whose centres are in general position, which determine Ω(n3/2) 4-rich points.

Furthermore, we generalise this to give a construction of m n-pencils determining many m-rich points. Theorem 28. For any m ∈ N, there exist m n-pencils whose centres are in general position 3/2 which determine Ωm(n ) m-rich points.

For a precise version of this result with the dependence on m made explicit, see the forthcoming Proposition 1.

4.2 Connection with the Sum-Product Problem

The construction relating to Problem 1 given in [2] arose from some surprising constructions for the sum-product problem restricted to graphs. In addition to Conjecture 1, Erd˝osand Szemerédialso considered taking sums and products restricted to a specified subset of A × A, as follows. Let G be a bipartite graph with vertices being two distinct copies of A, and let E(G) ⊆ A × A be the edges of G. We define the sumset of A along G to be

A +G A = {a + b :(a, b) ∈ E(G)}.

In more generality, for A and B two finite subsets of R, we take a set of edges E(G) ⊆ A × B, and define the sum set A +G B = {a + b :(a, b) ∈ E(G)}. The restricted product set, ratio set etc. are defined in the same way. Erd˝osand Szemerédi also gave a stronger version of their conjecture in this restricted setting, essentially saying that for sufficiently dense graphs G ⊂ A × A, at least one of |A +G A| or |A ·G A| is close to |G|. In [2], the authors gave several constructions to show that this stronger conjecture, and variants thereof, do not hold. One such result was the following.

38 Theorem 29 (Alon, Ruzsa, Solymosi). For arbitrarily large n, there exists A ⊆ R ﬁnite with |A| = Θ(n), and a subset S ⊆ A × A with |S| = Ω(n3/2), such that S is the set of edges of a graph G with |A +G A| + |A/GA| = O(n).

Both the sumset and the ratio set are at most linear in size, but the graph has many edges. The construction used in this theorem is then converted, via a projective transformation, into a construction of a set of four n-pencils of lines, with non-collinear centres, that determine Ω(n3/2) 4-rich points. Similarly, our results in Theorems 5, 26, and 27 follow from considering sum-product type problems restricted to graphs. The sum-product problem that is most relevant to this paper is that of showing that if the product set of A is small, then the product set of a shift of A must be large. In this direction, it was proven by Garaev and Shen [33], that for any finite A, B, C ∈ R and any non-zero x ∈ R, max{|AB|, |(A + x)C|} |A|3/4|B|1/4|C|1/4. (4.1) This result and its proof closely follow the seminal work of Elekes [24] in which the Szemerédi- Trotter Theorem was first used to prove sum-product results, as shown in Chapter 1. In the process of proving Theorems 5, 26, and 27, we obtain some results about this version of the sum-product problem restricted to graphs which may be of independent interest. For example, we prove the following result. Theorem 30. For arbitrarily large n, there exists A, B ⊆ Q with |A|, |B| n, and a subset 3/2 43 S ⊆ A × B with |S| = Ω(n log(n) 1000 ), such that S is the set of edges of a graph G with

|A/GB| + |(A + 1)/GB| + |(A + 2)/GB| n.

In the above A/GB := {a/b :(a, b) ∈ E(G)}. More generally, for any x, y ∈ R, a + x (A + x)/ (B + y) := :(a, b) ∈ E(G) . G b + y

4.3 Proof of Theorem 5

We begin by giving a way to translate a question concerning pencils into a question concerning ratio and sum sets. The setup here is similar to that of Chang and Solymosi [16].

We take four non-collinear pencils L1, L2, L3, and L4, with |Li| = n for each i. As they are non-collinear, there exists a pair (say L1 and L2) such that the line connecting the centres of these pencils does not contain the centre of L3 or L4. We apply a projective transformation to send the centres of L1 and L2 to the projective coordinates (1; 0; 0) and (0; 1; 0) respectively. L1 now consists of horizontal lines, and L2 of vertical lines. By the choice we made, both the pencils L3 and L4 have aﬃne centres.

Pencils L1 and L2 deﬁne a Cartesian product A × B, where |A|, |B| = n. Let S ⊆ A × B be the set of 4-rich points. Let (x1, y1) and (x2, y2) be the centres of L3 and L4 respectively. Both L3 and L4 cover S, and by identifying an element λ of (B − y1)/G(A − x1) with its corresponding line of slope λ through (x1, y1), we have

(B − y1)/G(A − x1) ⊆ L3 =⇒ |(B − y1)/G(A − x1)| ≤ n (4.2)

39 L1

L3 L4

Figure 4.1: An example of four pencils after a projective transformation.

(B − y2)/G(A − x2) ⊆ L4 =⇒ |(B − y2)/G(A − x2)| ≤ n, where G is the bipartite graph on A × B induced by taking the set of edges to be S. We see that the question now concerns bounding S, the amount of edges of the graph G. We prove the following lemma, which is based on the proof of inequality (4.1) given in [33].

Lemma 5. Let A, B be ﬁnite sets of real numbers, and let |A| = |B| = n. Let (x1, y1), (x2, y2) be two distinct points in R2, and let G be a bipartite graph on A × B . Then |E(G)|3/2 |(B − y )/ (A − x )| + |(B − y )/ (A − x )| . 1 G 1 2 G 2 n7/4

Proof. Since the points (x1, y1) and (x2, y2) are distinct, at least one of x1 6= x2 or y1 6= y2 holds. We will assume without loss of generality that y1 6= y2. We also assume, without loss of generality, that x1, x2 ∈/ A, so as to avoid issues with division by zero. Furthermore, we can assume that |E(G)| ≥ Cn3/2 for some suﬃciently large constant C, as otherwise the result holds for trivial reasons. Indeed, for any x1 ∈ R \ A, y1 ∈ R and any graph G on A × B with |E(G)| n3/2, |E(G)| |E(G)|3/2 |(B − y )/ (A − x )| ≥ . 1 G 1 |A| |A|7/4

Let P = (B − y1)/G(A − x1) × (B − y2)/G(A − x2). Deﬁne the line la1,a2 by the equation

(a2 − x2)y = (a1 − x1)x + (y1 − y2), and let L = {la1,a2 : a1, a2 ∈ A}. Since, y1 6= y2, all of these 2 2 lines are distinct, and so |L| = |A| = n . For each b ∈ B, if (a1, b), (a2, b) ∈ E(G), the pair b−y1 b−y2 , ∈ P lies on line la ,a . For b ∈ B, let N(b) denote the neighbourhood of b in G, a1−x1 a2−x2 1 2 that is, N(b) := {a ∈ A :(a, b) ∈ E(G)}. Then we have a bound for the number of incidences: X I(P,L) ≥ |N(b)|2 b∈B |E(G)|2 ≥ n

40 by Cauchy-Schwarz. We use the Szemer´edi-Trotter theorem to bound on the other side as |E(G)|2 |P | + |L| + (|P ||L|)2/3. n Since |E(G)| ≥ Cn3/2 and |L| = n2, the middle term here can be dismissed and we have |E(G)|2 |P | + (|P ||L|)2/3. (4.3) n

If the second term on the right-hand side dominates, we get

h i2/3 |E(G)|2 |(B − y )/ (A − x )||(B − y )/ (A − x )| n4/3 , 1 G 1 2 G 2 n and so |E(G)|3/2 |(B − y )/ (A − x )| + |(B − y )/ (A − x )| . 1 G 1 2 G 2 n7/4 If, on the other hand, the ﬁrst term on the right hand side of (4.3) dominates, we get a stronger inequality than that claimed in the statement of the lemma.

Continuing with our four pencils from before, we had the information from the inequalities (4.2), which when we combine with Lemma 5 gives

|E(G)|3/2 n |(B − y )/ (A − x )| + |(B − y )/ (A − x )| 1 G 1 2 G 2 n7/4 so that the number of edges, and thus the number of four-rich points, satisﬁes

|E(G)| n11/6.

This concludes the proof of Theorem 5. This argument can be repeated to give similar results in other fields by using a suitable replacement for the Szemerédi-Trotter Theorem. In the complex setting we can use a result of Tóth[87] (see also Zahl [95]), obtaining the same results as above. Over Fp we can use an incidence theorem for Cartesian products due to Stevens and de Zeeuw [83]. We calculated 2− 1 that this gives an upper bound O(n 8 ) for the number of 4-rich points.

4.4 Proof of Theorem 26

In order to prove Theorem 26, we will ﬁrst prove Theorem 30. We will then show this sum- product construction implies a construction with four pencils determining many 4-rich points. We make use of the following theorem due to Ford [32] concerning the product set of the ﬁrst n integers.

Theorem 31. Let A(n) be the number of positive√ integers m ≤ n which can be written as a product m = m1m2, where m1, m2 ∈ {1, 2, ..., b nc}. Then n A(n) ∼ (log n)δ(log log n)3/2

1+log log 2 where δ = 1 − log 2 = 0.086071 ... .

41 As a corollary, we re-write this theorem in the language of product sets. Corollary 6. Let A = {1, 2, ..., n}. Then the product set AA has size n2 |AA| 43 . (log n) 500

Here we have absorbed the log log factor by slightly reducing the exponent of the log factor, for simplicity of the forthcoming calculations. We now have the tools to prove Theorem 30.

Proof of Theorem 30. Let d > 0 be some parameter to be chosen later. Deﬁne the sets √ √ i n n A = : i, j ∈ Z, (i, j) = 1, 1 ≤ i, j ≤ , j ≥ (4.4) j (log n)d 2(log n)d 1 n B = : l ∈ Z, 1 ≤ l ≤ . (4.5) l (log n)d Note that n |A| ∼ . (log n)2d Indeed, the number of coprime pairs of integers less than some parameter x is asymptotically 6 2 equal to π2 x , and so √ √ 6 n 2 6 n 2 n |A| ≥ − + lower order terms . π2 (log n)d π2 (2 log n)d (log n)2d

We deﬁne a bipartite graph on A×B, where the edges E(G) are deﬁned by the following. i 1 E(G) = , ∈ A × B : j|l . j l The number of edges is given by the formula √ X n n o n |E(G)| = 1 ≤ i ≤ :(i, j) = 1 k ∈ Z : 1 ≤ kj ≤ . d d √ √ (log n) (log n) n ≤j≤ n 2(log n)d (log n)d n o Z n n The size of the set k ∈ : 1 ≤ kj ≤ (log n)d gives the amount of multiples of j up to (log n)d . √ √ n n √ As 2(log n)d ≤ j ≤ (log n)d , the amount of these multiples is ∼ n. We can thus move this outside of the sum over j, obtaining √ √ n n o √ n3/2 X |E(G)| ≥ n 1 ≤ i ≤ d :(i, j) = 1 = n|A| 2d . √ √ (log n) (log n) n ≤j≤ n 2(log n)d (log n)d

The ratio set A/GB consists of the elements il i 1 A/ B = such that ∈ A, ∈ B, j|l G j j l √ n √ ⊆ il0 : 1 ≤ i ≤ , 1 ≤ l0 ≤ 2 n (log n)d ⊆ CC

42 √ where C = {1, 2, ..., 2 n}. Thus we have3 by Corollary 6 n |A/GB| 43 . (log n) 500

When we apply a shift of 1 to A and calculate the ratio set (A + 1)/GB, we get the same result. (i + j)l i 1 (A + 1)/ B = : ∈ A, ∈ B, j|l G j j l √ √ √ n n n √ ⊆ (i + j)l0 : 1 ≤ i ≤ , ≤ j ≤ , 1 ≤ l0 ≤ 2 n (log n)d 2(log n)d (log n)d √ 2 n √ ⊆ kl0 : 1 ≤ k ≤ , 1 ≤ l0 ≤ 2 n ⊆ CC. (log n)d

For (A + 2)/GB we ﬁnd an extra constant, but we still have the same result. We now have the sum n |A/GB| + |(A + 1)/GB| + |(A + 2)/GB| 43 (log n) 500 where the amount of edges on G is n3/2 |E(G)| . (log n)2d

43 n We now set d = , and let m = 43 . This gives us the following; 1000 (log n) 500 n |B| ≥ |A| 43 = m, (log n) 500 n |A/GB| + |(A + 1)/GB| + |(A + 2)/GB| 43 = m, (log n) 500 3/2 n 3/2 43 |E(G)| ≥ m (log m) 1000 , (log n)2d thus completing the proof.

We can immediately use this result to create a set of four pencils with many 4-rich points.

Proof of Theorem 26. We consider our construction from Theorem 30. The edges of the graph 2 correspond to a set S ⊆ A × B ⊂ R . The amount of elements of A/GB and the two shifts are exactly the amount of lines needed to cover S through either the origin for A/GB, the point (−1, 0) for (A + 1)/GB or (−2, 0) for (A + 2)/GB. These are our first three pencils, which we already know have cardinality O(m). Our fourth pencil will have its centre on the line at infinity, and will consist of vertical lines covering S. The amount needed is precisely |A| = O(m). The amount of 4-rich points is at least the size of S, since each pencil covers S. 3/2 43 Thus we have at least m (log m) 1000 4-rich points. Note also that the centres of the four pencils we have chosen are non-collinear. The point at infinity met by the line connecting (0, 0), (−1, 0) and (−2, 0) is not the equal to the point corresponding to the centre of the fourth pencil. 3It is possible to be more careful here, and use an analogue of Ford’s result for an asymmetric multiplication table, in order to make a saving in the exponent of the logarithmic factor in Theorem 30 and thus in turn Theorem 26. In order to simplify the calculations we do not pursue this improvement.

43 4.5 Constructions with Arbitrarily many Pencils

We give a construction of a set where the sum-set, ratio set, an additive shift of the ratio set, and the diﬀerence set are all linear when we restrict to a graph, where the graph has many edges. We also show using shifts of ratio sets that there are sets of m n-pencils of lines that 3/2 determine Ωm(n ) m-rich points. Theorem 32. For arbitrarily large n, there exists a set A with |A| = Θ(n), and a graph G on A × A with Ω(n3/2) edges, such that

|A +G A| + |A/GA| + |(A + 1)/G(A + 1)| + |A −G A| n.

Proof. Let i √ A := :(i, j) = 1, 1 ≤ i, j ≤ n j √ The size of A is the amount of coprime pairs from 1 to n; therefore |A| = Θ(n). We deﬁne a bipartite graph G with vertex set A × A and

i k √ E(G) = , : 1, ≤ i, j, k ≤ n, (i, j) = 1 = (k, j) . j j

With this deﬁnition, we have |E(G)| n3/2. Indeed, X √ |E(G)| = |{(i, k) : 1 ≤ i, k ≤ n, (i, j) = 1 = (k, j)}| √ 1≤j≤ n X √ = |{i : 1 ≤ i ≤ n, (i, j) = 1}|2, √ 1≤j≤ n and so by the Cauchy-Schwarz inequality,

 2 2 X √ n  |{i : 1 ≤ i ≤ n, (i, j) = 1}| √ 1≤j≤ n √ X √ √ ≤ n |{i : 1 ≤ i ≤ n, (i, j) = 1}|2 = n|E(G)|, √ 1≤j≤ n as claimed.

n i+k √ o • The sum set restricted to G is A +G A ⊆ : i, j, k ∈ [ n] . The numerator ranges √ j √ from 1 to 2 n, and the denominator from 1 to n, thus |A +G A| n.

i √ • The ratio set is A/GA ⊆ k : i, k ∈ [ n] = A, so |A/GA| n.

n i+j √ o • The shifted ratio set is (A + 1)/G(A + 1) ⊆ k+j : i, j, k ∈ [ n] and so |(A + 1)/G(A + 1)| n.

n i−k √ o • Finally, the diﬀerence set is A −G A ⊆ j : i, j, k ∈ [ n] , so |A −G A| n.

Therefore the sum of the sizes of these four sets is n.

44 Using the same construction, we may consider only ratio sets to generalise this to any number of pencils. We may arbitrarily shift the ratio set by any (x, y) ∈ Z2 and keep its size linear in n;

i + xj √ (A + x)/ (A + y) ⊆ : i, j, k ∈ [ n] G k + yj √ √ √ √ =⇒ |(A + x)/G(A + y)| ≤ ( n + x n)( n + y n) xyn, which gives a construction to prove the following proposition, a more precise version of Theorem 28.

Proposition 1. For any m ∈ N, there exists a set of m pencils of lines, with any three centres of pencils non-collinear, such that each pencil contains N lines, and the amount of m-rich points is Ω(N 3/2/m3).

Proof. To get the best possible dependence on m in this statement, we need to choose a set of m centres which are in general position, and so that their coordinates are as small as possible. It is possible to construct such a set of size m in the lattice [m] × [m]. We take P to be this set of centres.

Let A and G be deﬁned as above. Form (A + x)/G(A + y) for (x, y) ∈ P . The centres are non-collinear, each pencil contains m2n := N lines, and the amount of m-rich points is at least the amount of edges, thus Ω(n3/2) = Ω(N 3/2/m3).

Finally, note that by taking m = 4 in the previous proposition we obtain Theorem 27.

4.6 Expander Results from Pencils

In this section we apply ideas from this chapter to prove two expander results in arbitrary ﬁelds. We prove the following.

Theorem 33. For all A ⊆ F with |A| p1/2 in positive characteristic, we have a(b − 1) 6/5 : a, b ∈ A |A| . b

Theorem 34. For all A ⊆ F with |A| p1/2 in positive characteristic, we have a − b + 1 6/5 : a, b ∈ A |A| . a + b

As a by-product of the method, we also prove the following geometric result.

2 Theorem 35. Let l1, l2, l3, and l4 be four non-concurrent lines in F , and let L be a set of lines in F2. Then we have 1/2+1/30 max |li ∩ L| |L| . i

If F = R, the exponent can be improved to 1/2 + 1/22.

45 The threshhold bound for this problem over R is an exponent of 1/2, given by the Sze- S mer´edi- Trotter theorem applied to the point set P = i(L∩li) and the line set L. Furthermore, the presence of four lines is necessary; if only three lines were taken in the above then only an exponent of 1/2 may be reached, which is seen by the following construction.

2 Construction 1. There exists a set of three non-concurrent lines l1, l2, and l3 in F and a line set L such that 1/2 max |li ∩ L| |L| . i

Take the three lines l1, l2, and l3 to be the y-axis, x-axis, and the line at inﬁnity respectively. Let A ⊆ F be a geometric progression, and deﬁne the line set given by connecting 2 the points (0, a) and (b, 0) lying l1 and l2 respectively, for each pair (a, b) ∈ A . We have |l1 ∩ L| = |l2 ∩ L| = |A|, and |l3 ∩ L| is precisely the number of slopes present in L, which is |A/A| |A|. Therefore, 1/2 max |li ∩ L| |A| = |L| . i

Moreover, the restriction that the lines l1, l2, l3, and l4 are non-concurrent is necessary - otherwise we can take L to be a set of lines passing through the common point. A generalisation of Lemma 5 is the main tool in the proof of Theorem 35.

Lemma 6. Let A, B be ﬁnite subsets of F, with |A| = |B| = n. We assume n p1/2 in positive 2 characteristic p. Let (x1, y1), (x2, y2) be two distinct points in F , and let G be a bipartite graph on A × B . Then

|E(G)|8/5 |(B − y )/ (A − x )| + |(B − y )/ (A − x )| . 1 G 1 2 G 2 n2 Furthermore, |E(G)|8/5 |(B − y )/ (A − x )| + |B − x A| . 1 G 1 G 2 n2

The proof is the same as of Lemma 5 apart from an application of the Stevens - de Zeeuw incidence bound instead of Szemer´edi - Trotter. We give the proof for completeness.

Proof. Since the points (x1, y1) and (x2, y2) are distinct, at least one of x1 6= x2 or y1 6= y2 holds. We assume WLOG that y1 6= y2. We also assume WLOG, that x1, x2 ∈/ A, to avoid issues with division by zero. For simplicity, we denote (B−y1)/G(A−x1) := R, and (B−y2)/G(A−x2) := S. As in the prove of Lemma 5, we can assume that |E(G)| ≥ Cn3/2 for some suﬃciently large constant C, as otherwise the result holds for trivial reasons. Indeed, for any x1 ∈ F \ B, 3/2 y1 ∈ F and any graph G on A × B with |E(G)| n ,

|E(G)| |E(G)|8/5 |R| ≥ . |B| n2

Let P := R × S. Deﬁne the line la1,a2 by the equation (a2 − x2)y = (a1 − x1)x + (y1 − y2), 2 2 and let L = {la ,a : a1, a2 ∈ A}. Since y1 6= y2 these lines are distinct, and so |L| = |A| = n . 1 2 b−y1 b−y2 For each b ∈ B, if (a1, b), (a2, b) ∈ E(G), the pair , ∈ P lies on the line la ,a . For a1−x1 a2−x2 1 2

46 b ∈ B, let N(b) denote the neighbourhood of b in G, that is, N(b) := {a ∈ A :(a, b) ∈ E(G)}. Then we have a bound for the number of incidences: X I(P,L) ≥ |N(b)|2 b∈B |E(G)|2 ≥ n by Cauchy-Schwarz. We use the incidence theorem of Stevens and de Zeeuw [83] to bound on the other side. We restate the theorem here for convenience. Theorem. Let A, B ⊆ F be sets and L be a set of lines, with |A| ≤ |B| and |A||B|2 ≤ |L|3. In positive characteristic p we assume |A||L| ≤ p2. Then we have

I(A × B,L) |A|3/4|B|1/2|L|3/4 + |L|.

The latter two conditions needed to apply this theorem are certainly satisfied if we have max{|R||S|2, |R|2|S|} ≤ n6, which is trivially true, and n p1/2 in positive characteristic p, which is assumed. The final condition holds if we have |R| ≤ |S|, and if this is not true we may reflect the entire point-line system in the line y = x to swap the order of the Cartesian product. In particular, reflecting the system does not alter the incidences, and the two other conditions still hold. To take this possible reflection into account, we take the leading term given in each case, to find |E(G)|2 (|R|1/2|S|3/4 + |S|1/2|R|3/4)n3/2 + n2. n Since the result holds if we have |E(G)| n3/2, we may assume that the second term above does not dominate. Therefore the first term dominates, and after simplification we have

|E(G)|8/5 |R| + |S| n2 as needed. The second part of the lemma follows in the same way, altering the point set to

P = (B − y1)/G(A − x1) × (B −G x1A), and the line set to the set of lines la1,a2 given by y = (a1 − x1)x + (y1 − x2a2). For each line la ,a , and for each b with (b, a1) and (b, a2) edges in 1 2 b−y1 G, the point , b − x2a2 is incident to la ,a . The rest of the argument is identical. a1−x1 1 2

We prove the following corollary of Lemma 6.

2 Corollary 7. Let L1, L2, L3, and L4 be four pencils of lines in F with not all centres collinear, 1/2 such that |L1| ≤ |L2| ≤ |L3| ≤ |L4| and |Li| p in positive characteristic. Let P be the set of four-rich points deﬁned by these pencils. Then we have

|P |8/5 |L3| + |L4| 2 . |L2|

Proof. This proof is a more general version of the proof of the analogous result over R in this chapter. We begin with the four pencils of lines, order by index in terms of size. We take the smallest pencil, L1, and we add lines to it arbitrarily until we have |L1| = |L2|. We then project L1 to be centred at the vertical direction, and L2 to be centred at the horizontal direction. These two pencils now define a Cartesian product, which we shall denote A × B. The four rich points defined by the pencils now correspond to a subset of A × B, which is in correspondence with a bipartite graph G on A and B, i.e. (a, b) is a four rich point iff (a, b) ∈ E(G). There are

47 now two cases. Since not all centres of the pencils are collinear, we either have L3 and L4 both having aﬃne centres, or one of these two pencils is now centred on l∞. These two cases correspond to the two parts of Lemma 6. In both cases, applying Lemma 6 gives |P |8/5 |L3| + |L4| 2 |L2| as needed.

We now deduce Theorem 35 from Corollary 7.

Proof. We begin with four non-concurrent lines l1, l2, l3, and l4, and a set of lines L. We wish to show that at least one of the lines li is met by L in many places. We dualise this situation, mapping our lines li to points pi, and the set of lines L to a set of points P . Since the lines li were not concurrent, the points pi are not collinear. We claim that the minimum number of lines passing through pi needed to cover P is precisely |li ∩ L|. Indeed, multiple lines passing through a point q on the lines li are mapped to a set of points collinear with pi in the dual setting. Therefore each intersection point in li ∩ L corresponds to a unique line through pi covering at least one point of P . For each point pi, let Li denote the minimal pencil of lines through pi needed to cover P . We are now in the situation to use Corollary 7. We have four pencils Li, and the number of four rich points is, by deﬁnition, at least |P |. WLOG we assume the ordering of |Li| is as in the Corollary. We then have |P |8/5 |L|8/5 |L3| + |L4| 2 = 2 . (4.6) |L2| |L2| The result follows upon rearranging.

Finally, we prove Theorems 33 and 34.

Proof of Theorem 33. The result follows from an application of Corollary 7. WLOG we remove 0 from A if it is present. We let l1 be the y-axis, l2 the x-axis, l3 the line y = 1, and l4 the line x = 1. Note that the lines l3 and l4 are the reﬂections of each other in the line y = x. We deﬁne the line set L as follows: connect the two points (0, a) and (b, 0) lying on l1 and l2 respectively, for each pair (a, b) ∈ A × A. This gives the line set b L = y = b − x : a, b ∈ A . a This line set in also symmetric in the line y = x, and by considering the x-coordinate of the intersection points of l3 with L, we have a(b − 1) |l3 ∩ L| = |l4 ∩ L| = : a, b ∈ A . b We also have |l1 ∩ L| = |l2 ∩ L| = |A|. Applying Corollary 7 to the dual of this system, we have 16/5 a(b − 1) |A| 6/5 : a, b ∈ A |l3 ∩ L| + |l4 ∩ L| = |A| b |A|2 as needed.

48 Proof of Theorem 34. Let l1 be the line x = −1/2, l2 the line x = 1/2, l3 the line y = x and l4 the line y = −x. The lines l3 and l4 are reﬂections of each other in the y-axis. We deﬁne the line set L as follows: for each pair of points (−1/2, a) and (1/2, b) lying on l1 and l2 respectively, we draw the line connecting these two points for each pair (a, b) ∈ A2. This gives a set of |A|2 lines, which can be written as

a + b L = y = (b − a)x + : a, b ∈ A . 2

This line set is symmetric in the y-axis, and by considering the x-coordinate of the intersection points of l3 with L, we have a − b + 1 |l3 ∩ L| = |l4 ∩ L| = : a, b ∈ A . a + b Applying Corollary 7 to the dual of this system, we have

16 a − b + 1 |A| 5 6/5 : a, b ∈ A |l3 ∩ L| + |l4 ∩ L| = |A| a + b |A|2 as needed.

49 Chapter 5

Constructions for the Elekes - Szab´o and Elekes - R´onyai Problems

5.1 Preliminaries

In this chapter we explore constructions for the Elekes - Szabóand Elekes - Rónyai problems. We give a construction of a non-degenerate polynomial F ∈ R[x, y, z] and a set A of cardinality n such that F vanishes on Ω(n3/2) points of A × A × A, thus providing a new lower bound construction for the Elekes–Szabóproblem. We also give a related construction for the Elekes– Rónyai problem restricted to a subgraph. This consists of a polynomial f ∈ R[x, y] that is not additive or multiplicative, a set A of size n, and a subset P ⊂ A × A of size Ω(n3/2) on which f takes only n distinct values. The results in this chapter first appeared in [45].

5.1.1 The Elekes–Szab´oProblem

We define the zero set Z(F ) of a polynomial F ∈ R[x1, . . . , xn] as n Z(F ) = {(a1, . . . , an) ∈ R : F (a1, . . . , an) = 0}. In other words, Z(F ) is the set of points where the polynomial vanishes. Elekes and Szabó[29] considered the size of the intersection of the zero set of a polynomial F (x, y, z) ∈ R[x, y, z] of degree d with a Cartesian product A × B × C ⊂ R3, where |A| = |B| = |C| = n. By the Schwartz–Zippel Lemma (see for instance [54, Lemma A.4]), we have 2 |Z(F ) ∩ (A × B × C)| d n . (5.1) This bound cannot be improved in general. For example, if F (x, y, z) = x + y + z, A = B = {1, . . . , n}, and C = {−1,..., −n}, then |Z(F ) ∩ (A × B × C)| n2. More generally, if the equation F (x, y, z) = 0 is in some sense equivalent to an equation of the form ϕ1(x) + ϕ2(y) + 2 ϕ3(z) = 0, then we can choose A, B, C so that |Z(F ) ∩ (A × B × C)| n . The following definition makes this property precise. Definition 1. A polynomial F (x, y, z) ∈ R[x, y, z] is degenerate if there are a point v ∈ Z(F ) and open intervals I1,I2,I3 such that v ∈ I1 ×I2 ×I3, and for each i there is a smooth (infinitely differentiable) function ϕi : Ii → R with a smooth inverse, such that for all (x, y, z) ∈ I1 ×I2 ×I3 we have F (x, y, z) = 0 if and only if ϕ1(x) + ϕ2(y) + ϕ3(z) = 0.

50 Elekes and Szab´o[29] showed that if the polynomial is not degenerate in this sense, then the bound (5.1) can be improved to n2−η for some η > 0. A quantitative improvement to η = 1/6 was obtained by Raz, Sharir and de Zeeuw [54], leading to the following statement.

Theorem 36 ( [54]). Let F ∈ R[x, y, z] be a polynomial of degree d. If F is not degenerate, then for any A, B, C ⊂ R of size n we have

2−1/6 |Z(F ) ∩ (A × B × C)| d n .

Not much attention has been paid to lower bound constructions for this theorem. Elekes 2 2 [25]√ noted that for F = x + xy + y − z and A = {1, . . . , n} we have |Z(F ) ∩ (A × A × A)| n log n (actually, Elekes formulated this in a diﬀerent way, which we mention in the next section; see [20] for more discussion). This was the only known lower bound for Theorem 36, and some have suggested that the upper bound could be improved as far as O(n1+ε) for an arbitrarily small ε > 0; for instance, the fourth author wrote this in [20]. The main purpose of this paper is to show by means of a simple example that this is not the case, and that in fact the bound in Theorem 36 cannot be improved beyond O(n3/2). Our main result is the following theorem.

Theorem 37. There exists a polynomial F ∈ R[x, y, z] of degree 2 that is not degenerate, such that for any n there is a set A ⊂ R of size n with

|Z(F ) ∩ (A × A × A)| n3/2.

In Section 5.4, we brieﬂy discuss possible extensions of this theorem to polynomials in more variables.

5.1.2 The Elekes–R´onyai Problem

Before the work of Elekes and Szabó[29], Elekes and Rónyai [28] considered the question of bounding the image of a polynomial f ∈ R[x, y] restricted to a Cartesian product, assuming that f does not have a certain special form, which is specified in the following definition.

Deﬁnition 2. A polynomial f(x, y) ∈ R[x, y] is additive if there are polynomials g, h, k ∈ R[t] such that f(x, y) = g(h(x)+k(y)), and it is multiplicative if there are polynomials g, h, k ∈ R[t] such that f(x, y) = g(h(x) · k(y)).

Elekes and R´onyai [28] proved that if f ∈ R[x, y] is not additive or multiplicative, then for every A, B ⊆ R with |A| = |B| = n the image |f(A, B)| is superlinear in n. The current state of the art for this problem is the following result of Raz, Sharir and Solymosi [56].

Theorem 38 ( [56]). Let f ∈ R[x, y] be a polynomial of degree d. If f is not additive or multiplicative, then for any A, B ⊂ R of size n we have

4/3 |f(A, B)| d n .

2 2 √ Elekes [25] noted that if f(x, y) = x + xy + y and A = {1, . . . , n}, then |f(A, A)| n2/ log n. This is the best known upper bound construction for Theorem 38, which suggests that we may have |f(A, B)| n2− for all positive . This conjecture is widely believed, see for

51 instance Elekes [25] or Matouˇsek[46, Section 4.1]. The construction that we give in the proof of Theorem 37 does not translate into a construction that disproves this conjecture. Nevertheless, we show that there is a polynomial that takes only a linear number of values on a certain large subset of the pairs in A×A. This approach is partly inspired by work of Alon, Ruzsa and Solymosi [2] concerning constructions for the sum-product problem along graphs. Let G be a bipartite graph on two copies of A with edge set E(G) ⊂ A×A. For a polyno- 0 0 mial f ∈ R[x, y] we deﬁne the image of f along G to be fG(A, A) = {f(a, a ):(a, a ) ∈ E(G)}. Our result is the following.

Theorem 39. There exist a polynomial f ∈ R[x, y] of degree 2 that is not additive or multiplicative, a ﬁnite set A ⊂ R of size n, and a bipartite graph G on A × A, such that

3/2 |E(G)| n and |fG(A, A)| ≤ n.

5.2 Construction

In this section we prove Theorem 37. Deﬁne the polynomial F (x, y, z) = (x − y)2 + x − z. We set A = {1, . . . , n} and we consider the intersection of Z(F ) with A × A × A. Consider the subset √ T = (k, k + `, k + `2): k, ` ∈ Z, 1 ≤ k ≤ n/2, 1 ≤ ` ≤ n/2 ⊂ A × A × A.

Each choice of k and ` determines a distinct triple in T , so we have |T | n3/2. For each triple in T , we have F (k, k + `, k + `2) = (k − (k + `))2 + k − (k + `2) = 0, so that T ⊂ Z(F ). Therefore we have

|Z(F ) ∩ (A × A × A)| n3/2.

It remains to show that F is not degenerate in the sense of Definition 1. We will use an idea introduced by Elekes and Rónyai [28], which is that non-degeneracy can be verified using the following straightforward derivative test; see for instance [29, Lemma 33] or [20, Lemma 2.2].

2 2 Lemma 7. Let f : R → R be a smooth function on some open set U ⊂ R with fx and fy not identically zero. If there exist smooth functions ψ, ϕ1, ϕ2 on U such that

f(x, y) = ψ(ϕ1(x) + ϕ2(y)), then ∂2 (log |f /f |) x y (5.2) ∂x∂y is identically zero on U.

52 Suppose that F (x, y, z) = (x − y)2 + x − z is degenerate. Then there is some open neighborhood I1 × I2 × I3 intersecting Z(F ), such that F (x, y, z) = 0 if and only if ϕ1(x) + ϕ2(y) + ϕ3(z) = 0, for some smooth functions ϕ1, ϕ2 and ϕ3 with smooth inverses. Then, since −1 ϕ3 has a smooth inverse on I3, we can write ψ(t) = ϕ3 (−t), so that F (x, y, z) = 0 is equivalent 2 to z = ψ(ϕ1(x) + ϕ2(y)). At the same time, F (x, y, z) = 0 rewrites to z = (x − y) + x, so there is an open set U ⊂ I1 × I2 × I3 on which we have

2 ψ(ϕ1(x) + ϕ2(y)) = (x − y) + x.

We now check if the expression (5.2) for f(x, y) = (x − y)2 + x is identically zero on U. We have 2(x − y) + 1 log |fx/fy| = log = log |2x − 2y + 1| − log |2x − 2y|, −2(x − y) so ∂2 (log |f /f |) ∂ −1 1 1 1 x y = + = − . ∂x∂y ∂x x − y + 1/2 x − y (x − y + 1/2)2 (x − y)2 This expression equals zero only when y −x = 1/4, so it does not vanish on any nontrivial open set. Thus (5.2) is not identically zero on U, and by Lemma 7 this contradicts our assumption that F is degenerate.

5.3 The Elekes–R´onyai Problem along a Graph

We now prove Theorem 39, concerning the image of a polynomial along a subset of a Cartesian product. Deﬁne the polynomial f(x, y) = (x − y)2 + x. Set A = {1, . . . , n} and let G be the bipartite graph on A × A with the edge set √ E(G) = (k, k + `): k, ` ∈ Z, 0 ≤ k ≤ n/2, 0 ≤ ` ≤ n/2 ⊂ A × A.

We have |E(G)| n3/2. Applying f along any edge gives a non-negative integer

f(k, k + `) = (k − (k + `))2 + k < n.

This shows that |fG(A, A)| ≤ n.

It remains to prove that f is not additive or multiplicative. We could again do this using Lemma 7, but here we can use a more elementary approach. We treat the two cases separately. Additive case: Suppose f(x, y) = g(h(x) + k(y)). Note that g, h and k must have degree at most 2. We cannot have deg(g) = 1, since then f(x, y) would not have any cross term xy. If deg(g) = 2, then deg(h) = deg(k) = 1. We can write

2 g(t) = a2t + a1t + a0, h(x) = b1x + b0, k(y) = c1y + c0, with b1 and c1 non-zero. Then we have

2 2 f(x, y) = (x − y) + x = a2(b1x + b0 + c1y + c0) + a1(b1x + b0 + c1y + c0) + a0. (5.3)

53 Calculating the coeﬃcient for the y term on the right hand side and comparing with the left hand side, it follows that 2a2(b0 + c0) + a1 = 0. (5.4) On the other hand, calculating the coeﬃcient for the x term on the right hand side of (5.3) and comparing with the left hand side, it follows that

b1(2a2(b0 + c0) + a1) = 1. This contradicts (5.4). Multiplicative case: Suppose f(x, y) = g(h(x) · k(y)). We cannot have deg(g) = 2, since then h or k would have to be constant, and f(x, y) would not depend on both variables. Therefore we have deg(g) = 1. In this case, we must have deg(h) = deg(k) = 1. We can write

g(t) = a1t + a0, h(x) = b1x + b0, k(y) = c1y + c0 and 2 (x − y) + x = f(x, y) = a1((b1x + b0)(c1y + c0)) + a0. This is a contradiction, since there is no x2 or y2 term on the right hand side. This completes our proof that f is not additive or multiplicative, which completes our proof of Theorem 38.

5.4 Extensions to more Variables

5.4.1 Four Variables

One can consider the same problems for polynomials in more variables. Raz, Sharir and de Zeeuw [55] proved that for F ∈ R[x, y, s, t] of degree d and A, B, C, D ⊂ R of size n, we have

8/3 |Z(F ) ∩ (A × B × C × D)| d n , (5.5) unless F (x, y, s, t) = 0 is in a local sense (similar to Deﬁnition 1) equivalent to an equation of the form ϕ1(x) + ϕ2(y) + ϕ3(s) + ϕ4(t) = 0. A construction of Valtr [89] (see also [74, Section 5.3]) essentially shows that for

V (x, y, s, t) = (x − y)2 + s − t one can set A = B = {1, . . . , n2/3} and C = D = {1, . . . , n4/3}, so that

|Z(V ) ∩ (A × B × C × D)| n8/3.

This would show that (5.5) is tight, if it weren’t for the fact that A, B and C,D have diﬀerent sizes. (A similar, older, construction of Elekes [26, Example 1.16] achieves the same with the polynomial xy + s − t, but is less relevant to us here.) If we require that A, B, C, D have the same size (and then we may as well assume that they all equal A ∪ B ∪ C ∪ D), then we can take Valtr’s polynomial V (x, y, s, t) together with the set A = {1, . . . , n}. Similarly to in our proof of Theorem√ 37, considering quadruples of the form (k, k + `, m, m + `2) with 1 ≤ k, m ≤ n/2 and 1 ≤ ` ≤ n/2, we get

|Z(V ) ∩ (A × A × A × A)| n5/2.

54 It is not hard to verify (as in our proof of Theorem 37) that V (x, y, s, t) is not degenerate in the sense of [55], so this gives a lower bound construction for (5.5), which is the best known. Note that the polynomial F (x, y, z) in our proof of Theorem 37 can be obtained from Valtr’s polynomial V (x, y, s, t) by setting s = x and t = z.

5.4.2 More than four Variables

For more than four variables, we do not have a statement that is entirely analogous to Theorem 36 or (5.5). Bays and Breuillard [6] proved a similar statement for any number of variables, but without an explicit exponent, and with a different description of the exceptional form. Also, Raz and Tov [57] extended Theorem 38 to any number of variables, with an explicit exponent. Because for the Elekes–Szabóproblem in more than four variables we do not have explicit exponents, and also because the appropriate definition of degeneracy is not clear, we only briefly touch on constructions for more variables here. There are various ways of extending our constructions to more variables; one can for instance take the polynomial

2 F (x1, . . . , xm) = (x1 + ··· + xm−1) + x1 − xm and the grid Am, where A = [−n, 2n]. Consider the set

2 √ T = (k1, k2 − k1, . . . , km−2 − km−3, ` − km−2, k1 + ` ) : 0 ≤ ki ≤ n, 0 ≤ ` ≤ n .

Then we have T ⊂ Z(F ) ∩ Am, which implies

m m− 3 Z(F ) ∩ A n 2 .

This should be compared with the Schwartz–Zippel bound |Z(F ) ∩ Am| nm−1. A potential Elekes–Szab´otheorem in m variables, i.e. an explicit version of the result of Bays and Breuillard, m m−1−ηm would give a bound of the form |Z(F ) ∩ A | n for some ηm > 0, under the condition that F is not degenerate in some sense. Presuming that our polynomial F is not of this form, it would show that we must have ηm ≤ 1/2.

55 Chapter 6

Kakeya Sets and the Polynomial Method

6.1 Preliminaries and Deﬁnitions

Fn F Recall that a subset K ⊆ q of n-dimensional vectors over the finite field q of q elements is Fn called a Kakeya set in q if it contains a line in each direction. Using the polynomial method Fn n Dvir [21, Theorem 1.5] showed that any Kakeya set in q contains at least cnq elements with a constant cn depending only on n, see also [36, Theorem 2.11]. We prove analogues of Dvir’s result for non-degenerate conics, that is, parabolae and hyperbolae (but not ellipses which do not have a direction). We also study so-called conical Nikodym sets where a small variation of the proof provides a lower bound on their sizes. (Here ellipses are included.) The results in this chapter first appeared in [91] Fn Fn Fn A set N ⊆ q is called a Nikodym set in q if for each point x ∈ q there is a line L containing x such that L \{x} ⊆ N . A small variation of Dvir’s proof also provides that any Fn n Nikodym set in q contains at least cnq elements with a constant cn depending only on n, see [36, Theorem 2.9]. F2 Now let q be the power of an odd prime. The set of zeros (x, y) ∈ q of a polynomial 2 2 Q(X,Y ) = AX + BXY + CY + DX + EY + F, A, B, C, D, E, F ∈ Fq, F2 of degree 2, that is A, B and C are not all zero, is called a conic in q. In the degenerate case, that is Q(X,Y ) is reducible over the algebraic closure of Fq, we get a pair of (intersecting, parallel or identical) lines, a point or the empty set. We restrict ourselves to the non-degenerate case, that is, Q(X,Y ) is absolutely irreducible over Fq, since the degenerate case is either trivial or can be reduced to the previously studied case of a single line. We may assume C = 1, C = 0 and A = 1, or A = C = 0 and B = 1. After regular affine substitutions X aX + bY + e 7→ , ad 6= bc, Y cX + dY + f F∗ we are left with the following cases where g is any fixed non-square in q:

F2 −1 −1 F2 • A = C = 0, B = 1: hyperbola {(x, y) ∈ q : x 6= 0 and y = x } = {(t, t ) ∈ q : t ∈ F∗ q}.

56 F2 2 2 F • (A, C) ∈ {(1, 0), (0, 1)}: parabola {(x, y) ∈ q : y = mx } = {(t, mt ): t ∈ q}, where m ∈ {1, g}.

F2 2 2 F∗ • C = 1, A 6= 0: ellipse {(x, y) ∈ q : y = gx + k}, k ∈ q. (Note that conics deﬁned by Y 2 = X2 + k can be transformed into the form XY = 1 and are hyperbolae.)

For parabolae and hyperbolae the parametrisations

2 (x(t), y(t)) = (t, mt ), t ∈ Fq, and −1 F∗ (x(t), y(t)) = (t, t ), t ∈ q, respectively, are obvious. However, we can also derive parametrisations of ellipses (x(t), y(t)) q+1 where t ∈ Fq2 with t = 1, see Section 6.2 below. F2 To extend the deﬁnition of a conic to a general dimension n ≥ 2, we embed any conic in q Fn Fn into a plane in q . That is for some vectors a, b, c ∈ q where b and c are linearly independent:

Fn • A(n embedding of a) hyperbola in q is a set

−1 F∗ H = {a + tb + t c : t ∈ q}. (6.1)

Fn • A(n embedding of a) parabola in q is a set

2 P = {a + tb + t c : t ∈ Fq}. (6.2)

Fn • An (embedding of an) ellipse in q is a set

q+1 E = {a + x(t)b + y(t)c : t ∈ Fq2 , t = 1} (6.3)

F2 where (x(t), y(t))∈ q is given in Section 6.2.

(Without the linear independence of b and c the embedding can have fewer points than the embedded conic. Hence, a hyperbola has q − 1 points, a parabola q points and an ellipse q + 1 points.) We give adaptations of Dvir’s proof to give bounds on conical Kakeya and Nikodym sets defined as follows. Fn Fn A subset N ⊆ q is called a conical Nikodym set if for all x ∈ q there is a non-degenerate conic C of the form (6.1), (6.2) or (6.3) with x ∈ C and C\{x} ⊆ N . In order to define conical Kakeya sets, we must decide on how to define the ’direction’ of a conic which can be identified with the ’point(s) at infinity’ of the conic, that is, a hyperbola has two directions b and c, a parabola has one direction c, and an ellipse has no direction. Fn Fn A subset K ⊆ q is called a conical Kakeya set if for all d ∈ q \{0} Fn there exist a, b, c ∈ q such that b and c are linearly independent and there is a conic contained in K either of the form (6.1) with d ∈ {b, c} or of the form (6.2) with d = c. We prove the following Theorem.

57 Fn Theorem 40. Let S ⊆ q with n ≥ 2 be a conical Kakeya or Nikodym set, where q is a power of an odd prime. Then q − 1n |S| ≥ . 2n

n For conical Kakeya sets the lower bound cnq with a constant depending on n follows from [30, Corollary 1.10]. However, in contrast to [30] our constant is explicit and in Section 6.5 n we use the method of multiplicities of [22] to determine a constant of the form cn = c where c does not depend on n. F2 Moreover, at the end of the chapter we give an example of a subset of q of size q + 1 which contains for each c, resp., b an ellipse of form (6.3). Hence, it is necessary to exclude ellipses in the deﬁnition of conical Kakeya sets. In Section 6.2 we derive a parametrisation for ellipses needed in the proof of Theorem 40. In Section 6.3 we prove Theorem 40 for conical Nikodym sets and in Section 6.4 for conical Kakeya sets. In Section 6.5 we improve the constant cn using the method of multiplicities. In Section 6.6 we conclude with some ﬁnal remarks. For readers not familiar with the polynomial method we refer to the book of Guth [36] and the survey article of Tao [86] as excellent starting points.

6.2 Parametrisation of Ellipses

In this section we derive a parametrisation for ellipses, which is vital in our proof of Theorem 40 for elliptic Nikodym sets. F2 2 2 F∗ Consider the ellipse E = {(x, y) ∈ q : y = gx + k}, where g is a non-square in q and F∗ k ∈ q. By [41, Lemma 6.24] we have |E| = q + 1. (6.4) Using analogues s(t) and r(t) of sine and cosine for finite fields defined below, see for example [42, Definition 15.5], we are able to find parametrisations of ellipses. 2 F F F F2 Note that a solution z of z = g is not an element of q: z ∈ q2 \ q. Let (u, v) ∈ q be any fixed solution of v2 = gu2 + k which exists by (6.4). Then verify that s(t) = 2−1z(t − t−1), r(t) = 2−1(t + t−1), is a solution of s(t)2 = g(r(t)2 − 1). It can be easily checked that F∗ q+1 E = {(x(t), y(t)) : t ∈ q2 , t = 1} with x(t) = g−1vs(t) + ur(t), y(t) = us(t) + vr(t). q q q (q−1)/2 F∗ Since r(t) = r(t) and s(t) = s(t) (using z = zg = −z because g is a non-square in q) F2 F2 we have (r(t), s(t)) ∈ q, so that (x(t), y(t)) ∈ q.

58 6.3 Conical Nikodym Sets

In this section we prove Theorem 40 for conical Nikodym sets. Fn Proposition 2. Let N ⊆ q with n ≥ 2 and q the power of an odd prime be a conical Nikodym set. Then we have q − 1n |N | ≥ . 2n

q−1 n Proof. Suppose |N | < 2n . By [36, Lemma 2.4], there is a non-zero polynomial f with 1/n q−3 Fn f(s) = 0 for all s ∈ N , and deg(f) ≤ n|N | ≤ 2 . Take any x ∈ q . As N is conical Nikodym, there exists a conic C of the form (6.1), (6.2) or (6.3) with x ∈ C and C\{x} ⊆ N . We split into cases depending on the form of the conic C. Firstly assume the conic C is a parabola P. Parametrise this parabola as

2 P = {a + tb + t c : t ∈ Fq}. Applying these points to the polynomial f, we define F (t) = f(a + tb + t2c) a univariate polynomial in t of degree deg(F ) ≤ q − 3. We also know that it has q − 1 zeros corresponding to the points of the parabola lying in N , and thus must be zero on the whole parabola, in particular f(x) = 0. Secondly assume C is a hyperbola H. Parametrise this hyperbola as −1 F∗ H = {a + tb + t c : t ∈ q}. Applying these points to the polynomial f, we define F (t) = tdeg(f)f(a + tb + t−1c) a univariate polynomial in t of degree deg(F ) ≤ q − 3. We also know that it has q − 2 zeros corresponding to the points of the hyperbola H lying in N , and thus must be zero on the whole hyperbola. Again we find f(x) = 0. Thirdly we assume that C is an ellipse E. The number of points on this ellipse is q + 1. We use our parametrisation of an ellipse; it has form F∗ q+1 E = {a + bx(t) + cy(t): t ∈ q2 , t = 1} for some appropriate choice of a, b and c, and (x(t), y(t)) are given in Section 6.2. We consider the polynomial F (t) = tdeg(f)f(a + bx(t) + cy(t)). This polynomial is univariate in t of degree deg(F ) ≤ q − 3. We know that it has q zeros (in Fq2 ) corresponding to the points of the ellipse E lying in N , and thus must be zero on the whole ellipse. We again find that f(x) = 0. In all three cases we found that f(x) = 0. As x was chosen arbitrarily we conclude that Fn q−3 f(x) = 0 for all x ∈ q . As deg(f) ≤ 2 the polynomial f must be the zero polynomial, a contradiction.

6.4 Conical Kakeya Sets

In this section we prove Theorem 40 for conical Kakeya sets. Fn Proposition 3. Let K ⊆ q with n ≥ 2 and q the power of an odd prime be a conical Kakeya set. Then q − 1n |K| ≥ . 2n

59 q−1 n Proof. Suppose that |K| < 2n . By [36, Lemma 2.4] there exists f a non-zero polynomial q−3 with f(s) = 0 for all s ∈ K, with degree d ≤ 2 . We split this polynomial into a sum of its greatest degree part and the lower degree terms as

X i1 i2 in f = fd + g, deg(fd) = d, deg(g) < d, fd(x1, . . . , xn) = ei1,...,in x1 x2 ··· xn . i1+...+in=d

Fn Note that as fd is homogeneous, fd(0) = 0. Take any x ∈ q \{0}. As K is conical Kakeya, there exists some conic C of the form (6.1) or (6.2) with x appearing as c for parabolae and b or c for hyperbolae from Section 6.1. We split into two cases depending on which type of conic C deﬁnes. First assume C is a parabola P. It has parametrisation

2 P = {a + tb + t c : t ∈ Fq}.

We consider the polynomial F (t)=f(a + tb + t2c), which is univariate in t of degree 2d. Since f is zero on K, F (t) = 0 for all t ∈ Fq. Then as deg(F ) = 2d < q, F is identically zero. We 2d 2d 2 note that the coeﬃcient of t in F (t) is the coeﬃcient of t in fd(a + bt + ct ):

2 2 2 fd(a + bt + ct ) = fd(a1 + b1t + c1t , . . . , an + bnt + cnt )

X 2 i1 2 i2 2 in = ei1,...,in (a1 + b1t + c1t ) (a2 + b2t + c2t ) ··· (an + bnt + cnt ) .

i1+...+in=d

Upon multiplying out to ﬁnd the coeﬃcient of t2d we have

X i1 i2 in 2d 2d X i1 i2 in ei1,...,in c1 c2 ··· cn t = t ei1,...,in c1 c2 ··· cn i1+...+in=d i1+...+in=d 2d = t fd(c) and thus as F is identically zero, fd(c) = fd(x) = 0. Secondly we assume C is a hyperbola H. Up to the relabelling of t → t−1, we may assume it has parametrisation −1 F∗ H = {a + tb + t c : t ∈ q}. Consider the univariate polynomial F (t) = tdf(a + tb + t−1c), which is of degree deg(F ) = 2d < F∗ q − 1. The polynomial f vanishes on K, and so F (t) = 0 for all t ∈ q. As deg(F ) < q − 1 with F having at least q − 1 zeros, we have that F (t) is identically zero, in particular its constant term is zero. We calculate the constant term of F (t); it is precisely the coeﬃcient of t−d in −1 fd(a + bt + ct ),

−1 X −1 i1 −1 in fd(a + bt + ct ) = ei1,...,in (a1 + b1t + c1t ) ··· (an + bnt + cnt )

i1+...+in=d so the coeﬃcient of t−d is X i1 in ei1,...,in c1 ··· cn = fd(c). i1+...+in=d

Thus fd(c) = fd(x) = 0.

In both cases we have fd(x) = 0. Since we already knew that fd(0) = 0, we have fd(x) = 0 Fn for all x ∈ q . As d < q, fd is identically zero, which is a contradiction.

60 6.5 Improvements via the Method of Multiplicities

The ’method of multiplicities’ was used in [22], see also [86], to prove a constant of 2−n for line Kakeya sets. This involves Hasse derivatives and exploiting polynomials which vanish to a high multiplicity on a particular set.

n Let x = (x1, ..., xn) and f ∈ Fq[x]. For a vector i = (i1, . . . , in) ∈ N , the i’th Hasse derivative of f, which we denote f i(x), is the coeﬃcient of yi in the polynomial f(x + y), where i i1 i2 in y is the monomial y1 y2 ...yn . F Fn For f ∈ q[x] and a ∈ q , the multiplicity of f at a, denoted Mult(f, a), is the largest integer M such that for all vectors i ∈ Nn of weight wt(i) < M, the i’th Hasse derivative of f i is zero at a, that is, f (a) = 0, where wt(i) = i1 + ... + in. We make use of ﬁve results relating to multiplicities and Hasse derivatives. These results, with proofs, can be found in [22], see also [86]. Lemma 8. Hasse derivatives ’commute’ with taking homogeneous parts of highest degree. That is, for f ∈ Fq[x] of total degree d, letting fd denote the homogeneous part of f of degree d, we have i i (fd) (x) = (f )d0 (x) where d0 ≤ d − wt(i) is the degree of f i. Lemma 9. Taking i’th Hasse derivatives reduces multiplicity by at most the weight of i. That is, Mult(f i, a) ≥ Mult(f, a) − wt(i). Lemma 10. Multiplicities of compositions of polynomials f(g(x)) at a is at least the multiplicity of f at g(a). That is, Mult(f(g(x)), a) ≥ Mult(f(x), g(a)).

Lemma 11 (Vanishing lemma for multiplicities). Let f ∈ Fq[x] be of degree d. Then X Mult(f, a) > dqn−1 =⇒ f is the zero polynomial. Fn a∈ q Fn Lemma 12. Suppose S ⊆ q such that for some natural numbers m, d we have m + n − 1 d + n |S| < . n n

Then there is a non-zero polynomial f ∈ Fq[x] of degree at most d, such that Mult(f, s) ≥ m for all s ∈ S.

d n Note that Lemma 12 is satisﬁed if |S| ≤ m+n .

6.5.1 Conical Nikodym Sets

In this section we use the method of multiplicities to prove the following theorem. Fn Theorem 41. Let N ⊂ q be a conical Nikodym set, with q a power of an odd prime. Then we have 4 −n |N | ≥ 3 + qn = (3 + o(1))−nqn, q → ∞. q − 2

61 Proof. We begin by taking a large multiple of q, call it lq for some positive integer l, and deﬁne 4 m = 3 + l . q − 2

lq−1 n F Assume that |N | ≤ m+n . By Lemma 12, there is a non-zero polynomial f ∈ q[x] of degree d < lq, such that Mult(f, x) ≥ m for all x ∈ N . Let fd(x) be the homogeneous part of f of degree d, which we know is not the zero polynomial. We aim to show that fd has high Fn multiplicity everywhere in q , and thus must be the zero polynomial. Indeed, we will show it has multiplicity l everywhere. Nn Fn i Choose i ∈ with wt(i) < l, and z ∈ q . We aim to show that (fd) (z) = 0. The case z = 0 is trivial, so we assume z 6= 0. As N is conical Nikodym, there is a conic C such that z ∈ C and C\{z} ⊂ N . We split into cases depending on the conic C, aiming to show that f i(z) = 0. Case 1 - Parabola Assume C is a parabola, which we can parametrise as ct2 + bt + a. We know by the properties of f that Mult(f, ct2 + bt + a) ≥ m for q − 1 values of t. By Lemma 9, we have Mult(f i, ct2 + bt + a) ≥ m − wt(i). We can now use Lemma 10 to get

Mult(f i(cx2 + bx + a), t) ≥ Mult(f i, ct2 + bt + a) ≥ m − wt(i) for q − 1 values of t. Note that the polynomial f i(cx2 + bx + a) has degree d0 ≤ 2 deg(f i) ≤ 2(d − wt(i)). However, f i(cx2 + bx + a) has multiplicity at least m − wt(i) for q − 1 values of t, so that by Lemma 11, as d < lq, wt(i) < l, we have X Mult(f i(cx2 + bx + a), t) ≥ (q − 1)(m − wt(i))

t∈Fq > 2(d − wt(i)) ≥ deg(f i(cx2 + bx + a)) so that f i(cx2 + bx + a) is in fact the zero polynomial. But then f i(z) = 0, as needed. Case 2 - Hyperbola Assume C is a hyperbola, which we can parametrise as bt + ct−1 + a. We know by the properties of f that Mult(f, bt + ct−1 + a) ≥ m for q − 2 values of t. By Lemma 9, we have Mult(f i, bt + ct−1 + a) ≥ m − wt(i). We have d0 = deg(f i) ≤ d − wt(i), and we deﬁne the polynomial F (t) = td0 f i(bt + ct−1 + a) which has degree 2d0, and also has multiplicity at least m − wt(i) for q − 2 values of t. By the vanishing lemma, we have X Mult(F, t) ≥ (q − 2)(m − wt(i)) > 2(d − wt(i))

t∈Fq so that F (t) is the zero polynomial. In particular, when we input the value t0 6= 0 corresponding d0 i −1 d0 i to z on the hyperbola, we get zero. Then F (t0) = t0 f (bt0 + ct0 + a) = t0 f (z) = 0 =⇒ f i(z) = 0 as needed. Case 3 - Ellipse Assume C is an ellipse, which we can parametrise as bx(t) + cy(t) + a with b and c linearly independent. We know by the properties of f that Mult(f, bx(t)+cy(t)+a) ≥ m for q values of t.

62 By Lemma 9, we have Mult(f i, bx(t)+cy(t)+a) ≥ m−wt(i). We have d0 = deg(f i) ≤ d−wt(i), and we define the polynomial F (t) = td0 f i(bx(t) + cy(t) + a) which has degree 2d0, and also has multiplicity at least m − wt(i) for q values of t ∈ Fq2 . By the vanishing lemma, we have X Mult(F, t) ≥ q(m − wt(i)) > 2(d − wt(i)) F t∈ q2 so that F (t) is the zero polynomial. In particular, when we input the value t0 6= 0 corresponding d0 i d0 i to z on the ellipse, we get zero. Then F (t0) = t0 f (bx(t0) + cy(t0) + a) = t0 f (z) = 0 =⇒ f i(z) = 0 as needed. Fn This was for arbitrary z, so we have Mult(f, z) ≥ l for all z ∈ q , and we may use the vanishing lemma a final time to show X Mult(f, x) ≥ lqn > dqn−1 Fn x∈ q so f is in fact the zero polynomial, a contradiction. We may allow l to go to infinity, so that

lq − 1 n q − 1/l n q n |N | ≥ lim ≥ lim = l→∞ m + n l→∞ 3 + 4/(q − 2) + n/l 3 + 4/(q − 2) as needed.

6.5.2 Conical Kakeya Sets

In this section we adapt the proof of [22] for line Kakeya sets to conical Kakeya sets.

Fn Theorem 42. Let K ⊂ q be a conical Kakeya set, with odd q. Then we have q n |K| ≥ . 3

Proof. We begin by taking a large multiple of q, call it lq, and deﬁne m = 3l. lq−1 n F Assume that |K| ≤ m+n . By Lemma 12, there is a non-zero polynomial f ∈ q[x] of degree d < lq, such that Mult(f, k) ≥ m for all k ∈ K. Let fd denote the homogeneous part of f with highest degree d. We will show that this polynomial has multiplicity l everywhere, so that f must be the zero polynomial. Fn Nn Let c ∈ q be arbitrary and non-zero (the zero case is trivial), and take i ∈ with wt(i) < l. As K is conical Kakeya, there is either a parabola or a hyperbola with direction c i contained in K. We split into cases, with the aim to show (f )d(c) = 0. Case 1 - Parabola Assume there is a parabola of the form ct2 + bt + a contained in K. 2 We know by the properties of f that Mult(f, ct + bt + a) ≥ m for t ∈ Fq. By Lemma 9, we have Mult(f i, ct2 + bt + a) ≥ m − wt(i). We can now use Lemma 10 to get

Mult(f i(ct2 + bt + a), t) ≥ Mult(f i, ct2 + bt + a) ≥ m − wt(i).

Note that the polynomial f i(ct2 + bt + a) has degree d0 ≤ 2 deg(f i) ≤ 2(deg(f) − wt(i)) = i 2 2(d − wt(i)). However, f (ct + bt + a) has multiplicity at least m − wt(i) everywhere in Fq, so

63 that by Lemma 11, as d < lq, wt(i) < l, we have X Mult(f i(ct2 + bt + a), t) ≥ q(m − wt(i))

t∈Fq > 2(d − wt(i)) ≥ deg(f i(ct2 + bt + a)) so that f i(ct2 + bt + a) is in fact the zero polynomial. The next observation is crucial; the coeﬃcient of x2 deg(f i) in f i(ct2 + bt + a) is precisely i (f )d0 (c), as only this highest degree homogeneous part could reach the highest power of x. But then by Lemma 8, we have i i (fd) (c) = (f )d0 (c) = 0 as needed. Case 2 - Hyperbola Up to a relabelling of t → t−1, we may parametrise the hyperbola as ct+bt−1+a. As the polynomial f(x) has multiplicity m everywhere in K, Mult(f, ct+bt−1+a) ≥ F∗ i −1 F∗ 0 m for t ∈ q. We then have that Mult(f , ct+bt +a) ≥ m−wt(i) for t ∈ q. Let d denote the degree of f i(x). We have d0 ≤ d−wt(i), and we deﬁne the polynomial F (t) = td0 f i(ct+bt−1 +a). F∗ Note that F (t) has multiplicity at least m−wt(i) for all t ∈ q, so that by the vanishing lemma, X Mult(F, t) ≥ (q − 1)(m − wt(i)) > 2(d − wt(i)) ≥ 2d0 = deg(F ).

t∈Fq

Therefore F (t) is the zero polynomial. In particular, its highest degree term is zero. The 2d0 i i i coeﬃcient of t in F (t) is precisely (f )d0 (c). By Lemma 8, we have (f )d0 (c) = (fd) (c) = 0, as needed. Fn We now have that Mult(fd, a) ≥ l for all a ∈ q . We may now use Lemma 11 to show

X n n−1 Mult(fd, a) ≥ lq > dq Fn a∈ q

lq−1 n so that fd is the zero polynomial, a contradiction. We therefore must have |K| ≥ m+n . As l was an arbitrary large integer, we may allow l → ∞, so we have

lq − 1 n q − 1/l n q n |K| ≥ lim = lim = l→∞ m + n l→∞ 3 + n/l 3 as needed.

6.6 Final Remarks

• For line Kakeya sets Dvir gave a construction of size at most

21−nqn + O(qn−1),

see [72, Theorem 7]. This construction can easily be adjusted to conical Kakeya sets. However, we lose a factor 2. We explain this for parabolae. For hyperbolae and ellipses one can deal analogously. Since otherwise our result is trivial we assume n ≥ 3. For

64 any direction c = (c1, . . . , cn) 6= 0 we take b = (b1, . . . , bn) with b1 = 1 and bi = 0 for i = 2, . . . , n if cn 6= 0 and (b1, . . . , bn−1, 0) any vector which is linearly independent to c if cn = 0. We also take a = (a1, . . . , an) with an = 0. Then for cn = 0 the parabola 2 Fn−1 n−1 a + tb + t c lies in q × {0} which contains q points. For cn 6= 0 choose b1 = 1 and bi = 0 for i = 2, . . . , n and note that b and c are linearly independent. Choosing 2 −2 2 ai = ci (2cn) for i = 2, . . . , n − 1 we see that a + tb + t c is of the form (α1, . . . , αn) with 2 2 4 2 2 −1 2 αi +αn = ai +t ci +t cn = (t cn +ci(2cn) ) for i = 2, . . . , n−1 by the choice of ai. Hence, 2 the parabola lies in the set {(α1, . . . , αn): αi +αn is a square for i = 2, . . . , n−1 and αn 6= 0}. We have q − 1 choices for αn, q for α1 and (q + 1)/2 for each αi with i = 2, . . . , n − 1. Hence, the size of our conical Kakeya set is at most

q + 1n−2 q (q − 1) + qn−1 = 22−nqn + O(qn−1). 2

• In [8, Definition 6], the authors introduced Kakeya sets of degree r which coincide with line Kakeya sets if r = 1. For r = 2 this definition differs from our definition of parabolic Kakeya sets by the condition that b and c are allowed to be linearly dependent. If q−1 n+1 q ≡ 1 mod (r + 1), in Lemma 7 they also give constructions of size at most r+1 + 1 . For r = 2 the construction is

3 n ci 3 K = + t − t : c1, . . . , cn, t ∈ Fq , q ≡ 1 mod 3. 3 i=1 However, to satisfy the linear independence condition we have to add lines for the direc- 2 2 tions for which (c1, . . . , cn) and (c1, . . . , cn) are linearly dependent, that is, (c1, . . . , cn) ∈ n F∗ 2 {0, c} for some c ∈ q and we have to add O(q ) further vectors, that is, we have the upper bound q + 2n+1 + O(q2). 3 It is not diﬃcult to extend Theorem 42 to such Kakeya sets of degree r (with a linear in- q n dependence condition) giving the lower bound r+1 . (Without the linear independence condition we can get only a weaker lower bound since the polynomial curves may contain only dq/re points.)

• For line Nikodym sets a lower bound (1 − o(1))qn is given in [35] where the implied constant is independent of n but depends on the characteristic of Fq. F3 • Improved lower bounds on (line) Kakeya and Nikodym sets in q are given in [43]. In F3 q3 2 particular it is shown that a construction for Nikodym sets in q of size 4 + O(q ) cannot exist and Nikodym sets behave diﬀerently than Kakeya sets where we have such a construction, see our ﬁrst remark.

• Modular conics, in particular hyperbolae, are well-studied objects. For a survey on modular hyperbolae see [81].

• The proofs of the lower bounds for the size of finite field Kakeya and Nikodym sets were inspired by ideas from coding theory, see for example [36, Chapter 4] and [94], more precisely from decoding Reed-Muller codes. The crucial idea is that a single missing value of a polynomial (of sufficiently small degree) on a line can be recovered. Similarly one can design decoding algorithms using non-degenerate conics instead of lines, see [94, Lemma 2.6] for parabolae.

65 • The following example shows for ellipses we can neither take b nor c as a direction to deﬁne elliptic Kakeya sets and prove a lower bound of order of magnitude qn. We take n = 2, q ≡ 3 mod 4 and q ≥ 19. Note that q ≡ 3 mod 4 if and only if −1 is a non-square F∗ F∗ F∗ F∗ in q, that is, for any non-square g in q the element −g is a square in q and let r ∈ q 2 F2 −1 be a square-root of −g, r = −g. Moreover, verify that q \{0} = {(c1r , c2):(c1, c2) ∈ F2 q \{0}}. Then set

−1 −1 2 2 2 2 −1 F2 K = {x(−c2r , c1) + y(c1r , c2): y = −x + (c1 + c2) , (c1, c2) ∈ q \{0}}

2 2 q−1 2 which deﬁnes only one ellipse E = {(x, y): y = gx + 1} with q + 1 < 4 points 2 2 by [41, Lemma 6.24] and since q ≥ 19. Note that c1 + c2 = 0 with (c1, c2) 6= (0, 0) is not possible since −1 is a non-square in Fq for q ≡ 3 mod 4. This example explains why we did not include the case of ellipses into the deﬁnition of conical Kakeya sets.

66 Chapter 7

Open Problems and Further Research

In this chapter we give five open problems concerning the sum-product phenomenon and discrete geometry. The five problems chosen to be included in this section have varying levels of fame and (possible) difficulty, and are placed in no particular order.

7.0.1 The Weak Erd˝os- Szemer´ediConjecture

A natural first step towards tackling the full Erd˝os- Szemerédiconjecture is to consider the two extreme cases. The first extreme case is when |A + A| |A|, in which case Conjecture 1 states that |AA| |A|2− for all > 0. This problem was mentioned in Subsection 3.2.1, and is called the ”Few sums, many products” problem. As we have seen, this problem was solved by Elekes and Ruzsa in [27], where it was proved that |A|2 |A + A| ≤ K|A| ⇒ |AA| (7.1) K4 log |A| coming from the general result |A|6 |A + A|4|AA| . log |A| We see that with the condition |A + A| ≤ K|A|, the dependence on K given in (7.1) is good, allowing us to give reasonable information about |AA| even when |A + A| is not quite minimal. The other extreme case is the opposite; if |AA| ≤ K|A|, what can we say about |A + A|? This problem is called the ”Few products, many sums” problem, or the ”Weak Erd˝os- Szemerédi conjecture”. If K is very small, a paper of Chang [15] answers this question. Chang proved the following. |AA| ≤ K|A| =⇒ |A + A| |A|2−, provided that K = o(log |A|). The range for K was extended with the restriction A ⊂ Q by Chang [13], whose result had an exponential dependence on K, and these ideas were further built upon in an acclaimed paper of Bourgain and Chang [11], where the following was proved. Theorem 43 (Bourgain-Chang). For all > 0, there exists C() such that if A ⊆ Q is a finite set with |AA| ≤ K|A|, then |A|2− |A + A| ≥ . KC()

67 In fact, Bourgain and Chang prove a much stronger result, see the expositional paper [98] of Zhelezov. Theorem 43 gives a few products, many sums result with polynomial dependence on K. Returning to R, a standard application of the Szemer´edi-Trotter theorem (see Elekes’ argument in Chapter 1) yields the result |A|3/2 |AA| K|A| =⇒ |A + A| K which places the ’trivial’ threshold bound at an exponent of 3/2. Current progress over the real numbers is given by the work of Olmezov, Semchankau, and Shkredov [52], who proved that |A|8/5 |AA| ≤ K|A| =⇒ |A + A| & . K14 Improvement to this result would be a vital step towards the full conjecture, and would likely have knock-on eﬀects for other questions of interest.

7.0.2 Additive Structure of Squares

In this section we consider problems concerning the additive structure of square numbers. We begin with a conjecture of Rudin, [65] concerning to what extent a set of squares can intersect an arithmetic progression. Conjecture 2. Let A ⊂ N be a ﬁnite set of squares, and let P ⊂ N be an arithmetic progression. Then we have |A ∩ P | |P |1/2.

If true, this conjecture would be aymptotically tight, as is seen by taking A = {1, 22, 32, ..., N 2}, and P = [N 2]. Progress on this conjecture was made by Bombieri, Granville and Pintz [10], who proved that |A ∩ P | . |P |2/3. This was then improved by Bombieri and Zannier [96] to |A ∩ P | . |P |3/5. Morally, this conjecture states that square numbers cannot be additively structured. A second conjecture concerning square numbers is that of Ruzsa. Conjecture 3. Let A ⊂ N be a ﬁnite set of squares. Then for all > 0, we have |A + A| |A|2−.

Note that this conjecture has the same moral idea behind it as Conjecture 2, however in contrast to Conjecture 2, this conjecture has proved to be exceptionally elusive, with no result of the form |A + A| |A|1+δ known. It is well known that Ruzsa’s conjecture and Rudin’s conjecture are linked, and we give an explanation of this in the next proposition. Proposition 4. A non-trivial result for Conjecture 2 implies a non-trivial result for Conjecture 3, and vice versa.

To show this, we use a central result in additive combinatorics known as Freiman’s The- orem. Loosely speaking, it says that if a set A ⊂ Z has small sumset, then A ﬁts eﬃciently inside a generalised arithmetic progression (GAP). A GAP P is a set of the form

P = {a1 + a2 + ... + ad : a1, a2, ..., ad ∈ Pi}

68 where P1, ..., Pd ⊂ Z are arithmetic progressions. The number d is called the rank, or dimension, Q Q of the GAP, and the value i |Pi| is its size. A GAP is called proper if |P | = i |Pi|. The following variant of Freiman’s theorem is due to Schoen [73], as is the argument proceeding it.

Theorem 44. Let A be a ﬁnite set of integers such that |A + A| K|A|. Then there exists √ C a proper generalised arithmetic progression P of dimension at most K log K and size at most √ C 48K24|A|, such that |A ∩ P | ≥ exp(−K log K )|A|.

We use a simple argument to show that, for example, the result of Bombieri and Zannier implies a non-trivial result of the form |A + A| = ω(|A|) for Conjecture 3. Let A be a set of squares, and let |A + A| ≤ K|A|. Then by Theorem 44, there exists a proper GAP of dimension and size as given in Theorem 44, which has a large intersection with A. Since we Qd have |P | = i=1 |Pi|, by the pigeonhole principle there is some arithmetic progression Pt with 1/d |Pt| ≥ |P | . WLOG we assume t = 1. We have X |A ∩ P | ≤ |A ∩ (P1 + x)|.

x∈P2+...+Pd

Since P1 + x is an arithmetic progression, we can apply the result of Bombieri and Zannier, 3/5 2/3 giving |A ∩ (P1 + x)| . |P1| |P1| . Here we have weakened logarithmic factors into a larger exponent for exposition, and to point out that the exponent here, as long as it is less than one, will make no diﬀerence to the ﬁnal result. We then have

X X |P1| |P | 1 2/3 1− 3d |A ∩ P | |P1| = 1/3 = 1/3 |P | . |P1| |P1| x∈P2+...+Pd x∈P2+...+Pd

Applying the lower bound for |A ∩ P | and the upper bound for |P |, after some simpliﬁcation we arrive at 1 24 √ C |A| 3d K exp(K log K ). Applying logarithms, we ﬁnd

log |A| √ C log K + K log K . 3d

√ C Multiplying through by d and then applying d ≤ K log K , we have

√ C √ 2C log |A| K log K log K + K log K .

Both terms on the right hand side here lead to the same result. Taking for example the second term, we find √ 2C c log log |A| log |A| K log K =⇒ K (log |A|) for some absolute c > 0. If the first term dominates, we either find a result of the same form, or a much stronger result. It is far easier to show the other direction: Assume we knew that for any A a set of squares, we have |A + A| |A|1+δ. We then have for any set A of squares, and any arithmetic progression P , |A ∩ P |1+δ |(A ∩ P ) + (A ∩ P )| ≤ |P + P | |P |

1 from which we conclude that |A ∩ P | |P | 1+δ . It was not necessary for P to be an arithmetic progression here; any set with small sumset would have suﬃced. This one line proof shows that a result of |A + A| |A|2− for Ruzsa’s conjecture would give a result of |A ∩ P | |P |1/2+

69 for Rudin’s conjecture. Because of this, Ruzsa’s conjecture can be regarded as stronger than Rudin’s conjecture. There is a third conjecture that should be mentioned here, due to Chang [14], concerning the additive energy of sets of squares.

Conjecture 4. Let A ⊂ N be a ﬁnite set of squares. Then for all > 0, we have

E+(A) |A|2+.

From the standard Cauchy-Schwarz inequality

|A|4 E+(A) |A + A| we see that this conjecture is stronger than Ruzsa’s conjecture. In [14], Chang proved that

|A|3 E+(A) . (log |A|)1/12

To summarise, the three conjectures above are all statements of the form ”square numbers cannot be additively structured”, each of diﬀerent strength. Although we have moderately good results for the ﬁrst conjecture, the lack of progress on the second and third conjectures is striking. Any progress on Conjecture 3, or even better Conjecture 4, would be of great interest. Some research has focused on giving progress for these conjecture while assuming the Bombieri-Lang conjecture, which is a deep conjecture in arithmetic geometry. For such results, see for instance [80] and [18], the second of which is also excellent further reading relating to problems in this section.

F2 7.0.3 Collinear Triples in p

In this section we consider the problem of upper bounding the number of collinear triples present in a Cartesian product A × A, for A ⊆ Fq, where A is not too large with respect to 3 the characterisic p. Formally, we look for the number of triples (p1, p2, p3) ∈ (A × A) which lie on a common line. Bounds on the number of collinear triples, and indeed collinear quadruples, over both R and Fp have been used to prove sum-product type results in many papers, see for instance [49] and [58]. Collinear quadruples were indeed used in the proof of Theorem 15 in Chapter 3. The analagous question over the real numbers is answered optimally (possibly up to logarithmic factors) by the Szemer´edi-Trotter theorem. That is, if we let

3 T (A) = {(p1, p2, p3) ∈ (A × A) : p1, p2, p3 lie on a common line} then we have the following |A|4 T (A) |A|4 log |A|. The lower bound is seen by considering axis parallel lines; there are |A| vertical lines deﬁned by A × A, and there are |A|3 triples on each such line. For the upper bound, we use the following version of the Szemer´edi-Trotter theorem.

70 2 Theorem 45. Let P ⊆ R be a ﬁnite set of points, and let Lk be the set of lines containing at least k points from P . Then we have

|P |2 |P | |L | + . k k3 k

We set P := A × A, and let L(P ) be the set of lines deﬁned by P . We count the number of collinear triples by the following sum

log |A| log |A| X 3 X X 3i X 3i T (A) = |l ∩ P | 2 ≤ |L2i |2 . l∈L(P ) i=1 l∈L(P ): i=1 2i≤|l∩P |<2i+1

Applying Theorem 45, we have

log |A| X |P | T (A) |P |2 + |A|4 log |A| + |A|4 |A|4 log |A|. 22i i=1

This bound does not hold in general in finite fields, since we could take A = Fp, and the total F2 5 5 number of collinear triples in p is Ω(p ). However, bounds better than |A| can be proved assuming that A is not too large with respect to the characteristic p, by making use of the point plane incidence theorem of Rudnev [67], which we restate here for convenience. Theorem (Rudnev). Let F be a field, and let P and Π be finite sets of points and planes respectively in P3. Suppose that |P | ≤ |Π|, and that |P | p2 if the characteristic p 6= 2 of F is positive. Let k be the maximum number of collinear points in P . Then the number of incidences satisfies I(P, Π) |Π||P |1/2 + k|Π|.

We give a proof of the following well known result.

2/3 Proposition 5. Let A ⊆ Fp be a ﬁnite set of points with |A| p . Then the number of collinear triples T (A) satisﬁes T (A) |A|9/2.

Proof. In order to apply the point-plane incidence theorem, we exploit the fact that three distinct points p1, p2, p3 are collinear if and only if the slope between p1 and p2 is the same as 4 the slope between p1 and p3. Note that there are |A| collinear triples coming from vertical lines. For triples not coming from vertical lines, we may write the slope condition algebraically, as a4 − a2 a6 − a2 ((a1, a2), (a3, a4), (a5, a6)) are collinear ⇐⇒ = . (7.2) a3 − a1 a5 − a1 After rearranging this equation, we reach

a5(a4 − a2) + a6(a1 − a3) = a1a4 − a2a3 which can be viewed as the point

(a4, a2, a5(a4 − a2)) lying on the plane Z + a6(a1 − a3) = a1X − a3Y.

71 Therefore, the number of collinear triples is at most the number of incidences between the set of points P = {(a4, a2, a5(a4 − a2)) : a2, a4, a5 ∈ A} and the set of planes

Π = {Z + a6(a1 − a3) = a1X − a3Y : a1, a3, a6 ∈ A}.

Note that we have |P | = |Π| = |A|3. To apply Rudnev’s theorem, we need that |A| p2/3, and we also need to know the maximum number of collinear points in P . We claim that the maximum number of collinear points in P is |A|, indeed any line not pointing in the z-direction can be projected to the first two coordinates, from which it is clear that the line can only contain |A| points from P . If the line is in the z-direction, then the first two coordinates from P are fixed, and then only the choice for a5 remains, giving |A| points. Applying Rudnev’s theorem, we find

I(P, Π) |A|9/2 + |A|4 |A|9/2 as needed.

F2 This result is the best known in p, although slightly improved bounds have been proved under further assumptions, such as small sumset or product set, in [49]. A general improvement of the form T (A) |A|9/2−c would be a very interesting result. The number of collinear triples can also be connected to the multiplicative energy of translates of A, as seen by counting solutions to equation in 7.2 by

X X ∗ r A−a1 (x)r A−a2 (x) ≤ |A| E (A − a). A−a1 A−a2 a1,a2∈A a∈A

This is another manifestation of the idea that multiplicative and additive structure cannot coexist - in this instance meaning that translation destroys multiplicative structure. It may be true that for all A ⊆ Fp suﬃciently small with respect to p, the number of values a ∈ A such that E∗(A − a) |A|3, is O(|A|) for all > 0. Such a result would lead to a near optimal result on collinear triples. One could even state the following conjecture.

Conjecture 5. For all A ⊆ Fp suﬃciently small with respect to p, and for all 0 ≤ δ ≤ 1, we have |{a ∈ A : E∗(A − a) |A|3−δ}| |A|δ.

Some restriction of the form ”A suﬃciently small with respect to p” must be present, by considering A = Fp. We also give a simple construction providing a lower bound.

1/4 1/2 Construction 2. For all δ ≥ 0, there exists A ⊆ Fp with p ≤ |A| ≤ p such that

|{a ∈ A : E∗(A − a) |A|3−δ}| |A|δ/3.

We put a lower bound on |A| into this statement since taking A to be a very small set, e.g. |A| 1, makes the statement trivial.

72 The construction consists of taking a union of many translates of a geometric progression. k 1−k Let P be a geometric progression in Fp of size N for some parameter N, and select N elements {x1, ..., xN k−1 } ∈ Fp. We then set

N k−1 [ A = ((P ∪ {0}) + xi). i=1

k We have that |A| ≤ N, and that A − xi contains a geometric progression of size N for all i = 1, ..., N 1−k. We therefore have

|{a ∈ A : E∗(A − a) |A|3k}| ≥ |{a ∈ A : E∗(A − a) N 3k}| ≥ N 1−k ≥ |A|1−k.

The result now follows from setting 3k = 3 − δ and rearranging, and noting that N 1/2 ≤ max{N k,N 1−k} ≤ |A| ≤ N, and setting N = p1/2.

7.0.4 Paley Graphs and Squares in Diﬀerence Sets

In this section we consider the following problem: How large can a set A ⊂ Fp be such that A − A contains only squares? The aim here is to gain knowledge on the distribution of squares in Fp - do they contain a large diﬀerence set? Note that we must restrict to the case where −1 is a square in Fp (i.e. p = 1 mod 4), since otherwise one of a − b, b − a ∈ A − A is a non-square.

This problem is often stated in terms of Paley graphs. For the finite field Fp with p = 1 mod 4, we have the associated Paley graph G(p) = G, with V (G) = Fp, and an edge between a, b ∈ Fp if the difference a − b is a square. Note that −1 being a square ensures that the edges are properly defined. Asking for large sets A such that A − A is a set of squares translates into asking for large cliques in G; the largest such A corresponds to the clique number ω(G). Finding upper bounds for ω(G), or equivalently for large sets A with A − A containing only squares, has proven to be very difficult. To provide context, we give a proof of the bound |A| ≤ (p − 1)1/2 + 1, using character sums.

Let χ be the quadratic character on Fp, and let us adopt the convention that χ(0) = 0. Suppose A ⊂ Fp is a set such that A − A contains only squares (and zero). We have X X |A|(|A| − 1) = χ(a − b) a∈A b∈A  21/2

1/2 X X ≤ |A|  χ(a − b)  a∈A b∈B  21/2

1/2 X X ≤ |A|  χ(x − b)  x∈Fp b∈B  1/2 1/2 X X x − b  = |A|  χ 0   0 x − b  x∈Fp b,b ∈B x6=b0 = |A|1/2(|A|(p − 1))1/2 which gives the result upon rearranging.

73 Lower bounds for ω(G) have been given in [19], where it is proved that ω(G) ≥ (1/2 + o(1)) log p, and then in [34], where this was improved to ω(G) log p log log log p. It was tentatively suggested (see [3]) that the correct order of magnitude is (log p)2. Unfortunately, despite the vast gap between (log p)2 and p1/2, no result of the form ω(G) = o(p1/2) has been reached. The current best result we have is due to Hanson and Petridis, [37], who used the polynomial method to show that ω(G) ≤ (p/2)1/2 + 1. It is interesting that applying the polynomial method, which tends to give strong results, still does not break into results of the form ω(G) = o(p1/2). This problem also has connections with finding the position of the first non-square in the vector (1, 2, ..., p − 1). If we had information that this first non-square was at position k, then the set A = {k/2, k/2 + 1, ..., k − 1} has only square differences (for simplicity we assume k even), and therefore ω(G) ≥ k/2.

7.0.5 Beck’s Theorem over Finite Fields

We recall from Chapter 1 the theorem of Beck. Let P ⊂ R2 be a ﬁnite set of points with at most k points collinear. Then we have

|L(P )| |P |(|P | − k).

Beck’s theorem has seen much use in combinatorial geometry, indeed it was used in Chapter 3 to prove Theorem 3. As mentioned in Chapter 3, the only obstacle preventing us from generalising F2 this result to p is the lack of a full strength Beck’s theorem in this setting. Having a strong Beck’s theorem in finite fields would, in addition to being a beautiful result in and of itself, give a powerful tool to prove geometric results in Fp by transferring the methods used over the reals. In the same way as many combinatorial results over Fp, a finite fields Beck’s theorem would necessarily have a condition on |P | with respect to the characteristic p of the field; as F2 2 usual, the counterexample is to take the whole plane p, which has p points, but defines only 2 p + p = O(|P |) lines. For large sets the following Beck’s theorem in finite fields Fq was proved by Alon [1].

F2 3 Theorem 46. Let P ⊂ q (q suﬃciently large) be a set of points with |P | ≥ 2 (q + 1). Then we have q2 |L(P )| ≥ . 24

For small sets a non-trivial Beck-type theorem can be proved using the incidence estimate of Stevens and de Zeeuw [83]. In their paper, they prove the following general incidence results.

F2 7/8 Theorem 47. Let P and L be sets of points and lines in p respectively, with |P | ≤ |L| ≤ 8/7 |L|13 15 |P | . Further, assume that |P |2 p . Then we have

I(P,L) |P |11/15|L|11/15.

74 Note that this theorem misses the 2/3 exponent present in Szemerédi- Trotter by 1/15. Theorem 47 is then used in [83] to prove the following finite field Beck’s theorem.

F2 7/6 Theorem 48. Let P ⊆ p be a set of points, with |P | p . Then one of the following is true.

1. Ω(|P |) points of P are collinear.

2. |L(P )| |P |8/7.

The exponent of 8/7 in the above theorem is too weak to make this theorem applicable. In practice, we often wish to apply Beck’s theorem to point sets with a specific structure - particularly Cartesian product. In this case, there is a stronger version of Beck’s theorem in finite fields, due to Yazici, Murphy, Rudnev, and Shkredov [93].

F2 2/3 Theorem 49. Let P = A × A be a set of points in p, with |A| < p . Then we have

|L(P )| |P |3/2.

In fact, we have essentially already given a proof of this theorem in the section on collinear triples; recall that we proved that if P = A × A, then the number of collinear triples deﬁned by P is at most |A|9/2. By H¨older’sinequality, we have

 2/3 4 X 2 1/3 X 3 1/3 2/3 |A| = |l ∩ P | ≤ |L(P )|  |l ∩ P |  = |L(P )| |T (A)| l∈L(P ) l∈L(P ) and thus an application of the collinear triples bound yields the result. This also shows that the possibly correct bound of T (A) . |A|4 would imply an optimal Beck’s theorem, up to logarithmic factors. Improvements to Beck’s theorem in ﬁnite ﬁelds for general point sets or for Cartesian products would be interesting, particularly if the strength of the result were good enough to give improvements to sum-product related problems.

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