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The Sum-Product Phenomenon and Discrete Geometry

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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, particu- larly 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 haupts¨achlich mit dem Summenprodukt-Ph¨anomen,bei dem es sich um das Prinzip handelt, dass sich eine endliche Teilmenge eines K¨orpers sowohl in Bezug auf Addition als auch in Bezug auf Multiplikation nicht gut verhalten kann (es sei denn, es ist nahe daran, ein Teilk¨orper zu sein). Von prim¨aremInteresse sind die reellen Zahlen und die endlichen K¨orper erster Ordnung. Diese Arbeit verwendet Ergebnisse aus diskreter Geometrie, um verschiedene Ergeb- nisse vom Summenprodukttyp zu verbessern. Verschiedene andere Ergebnisse in der kombi- natorischen / 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 - Szab´oand Elekes - R´onyai Problems 50 5.1 Preliminaries . 50 5.1.1 The Elekes{Szab´oProblem . 50 5.1.2 The Elekes{R´onyai Problem . 51 5.2 Construction . 52 5.3 The Elekes–R´onyai 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!1 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 f1; 2; 3; :::; ng 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 2 (1; 1) with 1=p + 1=q = 1, we have n n !1=p n !1=q X X p X q aibi ≤ jaij jbij : 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 multi- plication. 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 := fa + b : a; b 2 Ag AA := fab : a; b 2 Ag: 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 Szemer´edi[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 jAAj + jA + Aj ≥ cjAj2−: 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.
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