arXiv:1407.2397v2 [math.CO] 17 Aug 2014 aifistebound the satisfies ihsm plctost h inddsac rbe,a elproving well as problem, in distance circles pinned for the Theorem to applications some with praht iea siaeo niecso onsadshrsin spheres and points of incidences on we estimate Here an (1). give of proof to alternative an approach including fields, finite in problems analysis. Fourier discrete using proven be (1) ewe h onsadshrs hc sdntdby denoted is which spheres, and points the between htfrapitset point a for that hn2. than h eutwsetne oicdne ewe onsadhyperpla and points between incidences to extended was result The o xml 3 ,9 2.Frrltvl ag eso onsadsurfa and points of sets large relatively For 12]. 9, 6, [3, example for n ie epcieyin respectively lines and o xml,Vn 1]ue h pcrlmto opoeta,fo that, prove dea to to method tools spectral main the the used been [12] have Vinh theory example, graph For spectral and analysis AIRCLEUL,AE OEIH E UD LVRROCHE-N OLIVER LUND, BEN IOSEVICH, ALEX CILLERUELO, JAVIER LMNAYMTOSFRICDNEPOLM NFINITE IN PROBLEMS INCIDENCE FOR METHODS ELEMENTARY n[] h rtato on neeetr ehdt rv oer some prove to method elementary an found author first the [5], In ahmtc ujc lsicto 52C10 1 Classification Subject Mathematics rec in appeared have fields finite in problems incidence on results Many Let hnvrafiiefield finite a Whenever pee in spheres eemnsapstv rprino l ice.Telte euti an is constants. result multiplicative latter to up The optimal is circles. which all circles of for proportion Theorem positive a determines Abstract. F q eafil ihcaatrsi titygetrta 2. than greater strictly characteristic with field a be | P F q ||S| q n eueeeetr ehd opoea niec hoe o point for theorem incidence an prove to methods elementary use We sa plcto,w hwta n on set point any that show we application, an As . P | − F F q q 2 ⊂ P F . smnindi hsppr ti sue ohv hrceitcstrict characteristic have to assumed is it paper, this in mentioned is q 2 | 1 F , / q d 2 |S| I n family a and ( P, 1 / 2 L q ) d/ 1. ≤ 2 Introduction I < | FIELDS P RUDNEV ||L| q ( P, S 1 S ( + of ) < spheres | P | ||L| P I q ||S| ( P, q ) 1 S in + / := ) 2 . | 1 F P q d nti oe ti established is it note, this In P | h ubro incidences of number the , 1 |{ / 2 ⊂ ( sets r |S| ,S p, e in nes nlgeo Beck’s of analogue slso combinatorial on esults F ihteeproblems. these with l olwta elementary that follow WO N MISHA AND EWTON q 2 1 / ) 2 with F q eso fBeck’s of version a ∈ e in ces P q d d/ F P n lutaeit illustrate and 2 and . | q d P S × n er;see years; ent n a also can and , ≥ | and s F L q d : 5 Fourier , fpoints of p ygreater ly q ∈ S }| , 2 J. CILLERUELO, A. IOSEVICH, B. LUND, O. ROCHE-NEWTON ANDM. RUDNEV

In particular, the version of Beck’s Theorem for points and circles is tight up to multi- F2 plicative constants. In the forthcoming Theorem 3, it is established that any set P ⊂ q such that |P | ≥ 5q determines2 a positive proportion of all circles. On the other hand, if one takes a set P of q points lying on a single line in the plane, then P does not determine any circles. Similarly, if P consists of, say, q + 1 points on the same circle, then P determines only 1 circle. These degenerate examples are in a sense 1-dimensional, and illustrate the tightness of Theorem 3. Before saying any more about spheres in finite fields, it is necessary to define what is meant by such an object. We follow the notion of distance introduced in [7]; given a pair of Fd points x =(x1, ··· , xd) and y =(y1, ··· ,yd) in q, the distance between x and y is given by 2 2 ||x − y|| := (x1 − y1) + ··· +(xd − yd) . Fd As one might expect, a sphere in q is a set of points which are the same distance λ from a given central point (α1, ··· ,αd). That is, a sphere is the set of points x = (x1, ··· , xd) which satisfy an equation of the form 2 2 (x1 − α1) + ··· +(xd − αd) = λ. These spheres have many natural properties which are analogous with spheres in Rd. For 3 F2 example, given two circles in q, it is easy to check that the circles intersect in at most two points.

1.1. Work of Phoung, Thang and Vinh. Shortly after an earlier draft of this paper was made available online, we became aware of its overlap with a forthcoming paper of Phoung, Thang and Vinh [11]. The authors in [11] give an independent proof of Theorem 1 via graph theoretic methods similar to those used in [12]. Further applications of the incidence result are also given in [11].

2. Incidences between spheres and points

Given finite sets A, B in a finite group (G, +) we use the notation

rA+B(x)= |{(a, b) ∈ A × B : a + b = x}|. We recall the following elementary and well known identities:

(2) rA+B(x) = |A||B|, x∈G X 2 (3) rA+B(x) = rA−A(x)rB−B(x). x∈G x∈G X X The quantity in (3) is called the additive energy of A and B. 2 F2 Just as in the real plane, three noncollinear points in q determine a unique circle. A set of points P determines a given circles if three points from P determine that circle. More details of these basic properties of circles are given in section 3. 3 F2 Naturally, we call a sphere in q a circle. ELEMENTARY METHODS FOR INCIDENCE PROBLEMS IN FINITE FIELDS 3

Lemma 1. Define 2 2 F Fd+1 (4) A := {(a1, a2, ··· , ad, a1 + ··· + ad): a1, ··· , ad ∈ q} ⊂ q .

Then, for all x =(x1, ··· , xd+1) =6 (0, ··· , 0),

d−1 (5) rA−A(x) ≤ q .

Proof. This can be calculated directly. Indeed, the quantity rA−A(x) is the number of solu- tions F2d (a1, ··· , ad, b1, ··· , bd) ∈ q to the system of equations

a1 − b1 = x1

a2 − b2 = x2. . .

ad − bd = xd 2 2 2 2 a1 + ··· + ad − b1 −···− bd = xd+1.

The bi variables can be eliminated, and this system of equations reduces to 2 2 (6) 2a1x1 + ··· +2adxd − x1 −···− xd = xd+1.

If x =6 0 then there is some 1 ≤ i ≤ d such that xi =6 0. Otherwise xd+1 = 0 and x = 0 is the only choice which admits solutions to (6). Without loss of generality, we may take i = 1. If we fix a2, ··· , ad, then since the characteristic of the field is not equal to 2, we have d−1 2x1 =6 0 and the value of a1 is uniquely determined. This gives rA−A(x) ≤ q . d If x = 0, then trivially rA−A(0) = |A| = q . 

Fd+1 Lemma 2. Let A be as defined in (4), and let B,C ⊂ q be arbitrary. Then |B||C| |{(b, c) ∈ B × C : b − c ∈ A}| = + θ|B|1/2|C|1/2qd/2, q for some θ ∈ R such that |θ| < 1.

Proof. Note that |B||C| |C| |{(b, c) ∈ B × C : b − c ∈ A}| − = |{c ∈ C : b − c ∈ A}| − q q Xb∈B   |C| = r + (b) − := E. A C q Xb∈B   4 J. CILLERUELO, A. IOSEVICH, B. LUND, O. ROCHE-NEWTON ANDM. RUDNEV

By Cauchy-Schwarz: 2 2 2 |C| |C| |E| ≤|B| r + (b) − ≤|B| r + (x) − . A C q A C q b∈B Fd+1 X   x∈Xq   Using (2), (3) and the fact that |A| = qd we have

2 |C| 2 d−1 2 r + (x) − = r (x) − q |C| A C q A+C Fd+1 Fd+1 x∈Xq   x∈Xq d−1 2 = rA−A(x)rC−C (x) − q |C| Fd+1 x∈Xq d−1 d−1 2 ≤ |A||C| + q rC−C (x) − q |C| =0 Xx6 = |A||C| + qd−1(|C|2 −|C|) − qd−1|C|2 = |C|qd−1(q − 1). Thus, |E| < (|B||C|)1/2qd/2, which completes the proof.  Fd Fd Theorem 1. Let P ⊂ q and let S be a family of spheres in q . Then |P ||S| |P ||S| −|P |1/2|S|1/2qd/2

Proof. We denote by Sα1,··· ,αd,λ the sphere 2 2 {(x1, ··· , xd): (x1 − α1) + ··· +(xd − αd) = λ}. Define

B = {(p1, ··· ,pd, 0): (p1, ··· ,pd) ∈ P } and

C = {(α1, ··· ,αd, −λ): Sα1,··· ,αd,λ ∈ S}. Note that |B| = |P | and |C| = |S|. Now, note that |{(b, c) ∈ B × C : b − c ∈ A}|

= |{((p1, ··· ,pd, 0), (α1, ··· ,αd, −λ)) ∈ B × C :(p1 − α1, ··· ,pd − αd,λ) ∈ A}| 2 2 = |{((p1, ··· ,pd, 0), (α1, ··· ,αd, −λ)) ∈ B × C : (p1 − α1) + ··· +(pd − αd) = λ)}| = I(P, S). An application of Lemma 2 completes the proof.  ELEMENTARY METHODS FOR INCIDENCE PROBLEMS IN FINITE FIELDS 5

3. Applications of the incidence bound

Fd Fd 3.1. Pinned distances. Let P be a set of points in q, and y ∈ q. Following the notation of Chapman et. al. [4], let △y(P ) denote the set of distances between the point y and the set P ; that is

△y(P ) := {||x − y|| : x ∈ P }. It was established in ([4], Theorem 2.3) that a sufficiently large set of points determines many pinned distances, for many different pins. Here, we use Theorem 1 to give an alternative proof.

1 1 2 d+1 Fd − / 2 Corollary 1. Let P be a subset of q such that |P | ≥ ǫ (1 − ǫ) q for some 0 <ǫ< 1. Then, 1 (7) |△ (P )| > (1 − ǫ)q. |P | p Xp∈P

Proof. Fix a point p ∈ P , and construct a family of spheres Sp by minimally covering P by concentric spheres around p. Note that |Sp| = |△p(P )|, and that I(P, Sp)= |P |. Repeating this process for each point in P , we generate a family of spheres S defined by the disjoint union

S := Sp. p∈P 2 [ Observe that I(P, S)= p∈P I(P, Sp)= |P | . On the other hand, Theorem 1 implies that |P ||S| |P |2 = I(P, PS) < + |P |1/2|S|1/2qd/2 q 1/2 |P | |△p(P )| = p∈P + |P |1/2 |△ (P )| qd/2. q p P p∈P ! X 1 1 1 2 d+1 − / 2 If |P | p∈P |△p(P )| ≤ (1 − ǫ)q the last inequality would imply that |P | < ǫ (1 − ǫ) q .  P 2 2 1 2 d+1 Fd − / 2 Corollary 2. Let P be a subset of q such that |P | ≥ α (1 − α ) q for some 0 <α< 1. Then,

|△p(P )| > (1 − α)q for at least (1 − α)|P | points p ∈ P .

Proof. Corollary 1 implies that 2 |△p(P )| > (1 − α )q|P |. p∈P X On the other hand let ′ P = {p ∈ P : |△p(P )| ≥ (1 − α)q} 6 J. CILLERUELO, A. IOSEVICH, B. LUND, O. ROCHE-NEWTON ANDM. RUDNEV and suppose that |P ′| < (1 − α)|P |. Then we would have

|△p(P )| = |△p(P )| + |△p(P )| p∈P ′ ′ X p∈XP \P pX∈P < (1 − α)q(|P |−|P ′|)+ q|P ′| = (1 − α)q|P | + αq|P ′| < (1 − α2)q|P |. 

3.2. A version of Beck’s Theorem for circles. A result which is closely related to the Szemer´edi-Trotter Theorem and incidence geometry is Beck’s Theorem [2]. This result states that a set of N points in R2 determines Ω(N 2) distinct lines by connecting pairs of points, provided that the set of points does not contain a single line which supports cN points, where c is a small but fixed constant. We say that that P determines a line l if there exist two points p1 and p2 belonging to P which both lie on the line l. In finite fields, an analogue of Beck’s Theorem was proven by Alon [1], in the form of the following theorem: PF2 Theorem 2. Let ǫ > 0 and let P be a set of points in the projective plane q such that |P | > (1 + ǫ)(q + 1). Then P determines at least ǫ2(1 − ǫ) (q + 1)2 2+2ǫ distinct straight lines.

Iosevich, Rudnev and Zhai [8] used Fourier analytic techniques to establish a similar result. So, a sufficiently large set of points in the plane determines a positive proportion of all possible lines. The aim here is to establish an analogue of Theorem 2, with the role of lines replaced by circles. Since there are q2 choices for the location of a circle’s centre, and q choices for the radius, we want to generate Ω(q3) circles. We first need an obvious definition of what it means for a circle to be generated by a set of points. Given three non-collinear points in R2, there exists a unique circle which passes through F2 each of the three points. The same is true of three points in q: F2 Lemma 3. Let (x1,y1), (x2,y2) and (x3,y3) be three distinct non-collinear points in q. Then there exists a unique circle supporting the three points.

Proof. We’ll show that there exists a unique triple of elements a, b, c ∈ Fq such that the system of equations

2 2 (x1 − a) + (y1 − b) = c, 2 2 (8) (x2 − a) + (y2 − b) = c,  2 2  (x3 − a) + (y3 − b) = c,  ELEMENTARY METHODS FOR INCIDENCE PROBLEMS IN FINITE FIELDS 7 can be realised. Note that we cannot have x1 = x2 = x3, since this would contradict the hypothesis that the three points are non-collinear. Therefore, it is assumed without loss of generality that x1 =6 x2 and x1 =6 x3. Subtracting the first equation in (8) from the second and third yields the following system of linear equations:

2 2 2 2 2(x1 − x2)a + 2(y1 − y2)b = x1 + y1 − x2 − y2, (9) 2 2 2 2 2(x1 − x3)a + 2(y1 − y3)b = x + y − x − y .  1 1 3 3 It cannot be the case that both y2 = y1 and y3 = y1 (otherwise the three points would be collinear), and so at least one of the b coefficients is non-zero. Therefore, this is a system of two linear equations with two variables (a and b). This system has a unique solution (a, b), unless it is degenerate. This solution can then be plugged into (8) to give a unique solution to the system as required. It remains to show that (9) is non-degenerate. F∗ Suppose for a contradiction that (9) is degenerate. Then there exists λ ∈ q such that

λ(x2 − x1) = x3 − x1, (10) λ(y2 − y1) = y3 − y1,  Since it is known that at least one of y2 − y1 and y3 − y1 is non-zero, and λ is non-zero, it must be the case that both y2 − y1 =6 0 and y3 − y1 =6 0. Therefore,

x3 − x1 y3 − y1 λ = = . x2 − x1 y2 − y1 Hence y2 − y1 y3 − y1 =(x3 − x1) , x2 − x1 and clearly y2 − y1 y2 − y1 =(x2 − x1) . x2 − x1

This implies that (x1,y1), (x2,y2) and (x3,y3) are supported on a line with equation

y2 − y1 y = x + C, x2 − x1 where C = y1x2−x1y2 . This is a contradiction, and the proof is complete. x2−x1 

We say that a circle C is determined by P if there exist three points from P which determine the circle C. We are now ready to state our version of Beck’s Theorem for circles: F2 4q3 Theorem 3. Let P ⊂ q such that |P | ≥ 5q. Then P determines at least 9 distinct circles. 8 J. CILLERUELO, A. IOSEVICH, B. LUND, O. ROCHE-NEWTON ANDM. RUDNEV

Note that in the statement of Theorems 2 and 3, the conclusion is that we determine a positive proportion of all possible lines and circles respectively. If one asks how many points are needed to generate all lines (respectively circles), then the problem becomes rather F2 F2 different, since one can take a point set P = q \ l where l is a line (respectively P = q \ C where C is a circle), and the line l (respectively the circle C) is not determined by the point set P . So, we cannot hope to show that a set of o(q2) points determine all possible lines or circles.

3.3. Proof of Theorem 3. At the outset, identify a subset P ′ ⊂ P such that |P ′| = 5q. The aim is to show that P ′, and hence also P , determines many circles. Let S be the set of all circles which contain less than or equal to 3 points from P ′. We will show that

5q3 (11) |S| < , 9 3 4 3 and then since there are q circles, it must be the case that there are at least 9 q circles which contain at least 3 points from P ′. These 4q3/9 circles are therefore spanned by P . It remains to prove (11), and to do this we will make use of the lower bound on I(P ′, S) from Theorem 1. We have

|P ′||S| 5|S|−|P ′|1/2|S|1/2q = −|P ′|1/2|S|1/2q q

Arguments similar to the proof above can be found in [10]. Note that, using a lower bound F2 on the number of incidences between sets of points and lines in q which follows from the proof of the main result in [12], it is straightforward to adapt the proof of Theorem 3 with lines in the place of circles in order to prove a version of Theorem 2.

Acknowledgements. J. Cilleruelo was supported by grants MTM 2011-22851 of MICINN and ICMAT Severo Ochoa project SEV-2011-0087. Oliver Roche-Newton was supported by EPSRC Doctoral Prize Scheme (Grant Ref: EP/K503125/1) and by the Austrian Science Fund (FWF): Project F5511-N26, which is part of the Special Research Program “Quasi- Monte Carlo Methods: Theory and Applications. ELEMENTARY METHODS FOR INCIDENCE PROBLEMS IN FINITE FIELDS 9

References [1] N. Alon ‘Eigenvalues, geometric expanders, sorting in rounds, and Ramsey theory’, Combinatorica 6 (1986), no. 3, 207-219. [2] J. Beck, ‘On the lattice property of the plane and some problems of Dirac, Motzkin and Erd˝os in combinatorial geometry’, Combinatorica 3 (1983), no. 3-4, 281-297. [3] J. Bourgain, N. H. Katz and T. Tao, ‘A sum-product estimate in finite fields, and applications’, Geom. Funct. Anal. 14 (2004), no. 1, 27-57. [4] J. Chapman, M. Erdo˘gan, D. Hart, A. Iosevich and D.Koh, ‘Pinned distance sets, k-simplices, Wolff’s exponent in finite fields and sum-product estimates’, Math. Z. 271 (2012), no. 1-2, 63-93. [5] J. Cilleruelo, ‘Combinatorial problems in finite fields and Sidon sets’, Combinatorica 32 (2012), no.5, 497-511. [6] H. Helfgott and M. Rudnev, ‘An explicit incidence theorem in Fp’, Mathematika 57 (2011), no. 1, 135-145. [7] A. Iosevich and M. Rudnev, ‘Erd˝os distance problem in vector spaces over finite fields’, Trans. Amer. Math. Soc. 359 (2007), no. 12, 6127-6142. [8] A. Iosevich, M. Rudnev and Y. Zhai, ‘Areas of triangles and Beck’s theorem in planes over finite fields’, arXiv:1205.0107, (2012). [9] T. G. F. Jones, ‘Further improvements to incidence and Beck-type bounds over prime finite fields’, arXiv:1206.4517, (2012). [10] B. Lund and S. Saraf, ‘Incidence bounds for block designs’, arXiv:1407.7513, (2014). [11] N. D. Phoung, P. V. Thang and L. A. Vinh, ‘On the number of incidences between points and spheres in vector space over finite fields and related problems’, Forthcoming. [12] L. A. Vinh, ‘The Szemer´edi-Trotter type theorem and the sum-product estimate in finite fields’, Euro- pean J. Combin. 32 (2011), no. 8, 1177-1181.

J. Cilleruelo: Instituto de Ciencias Matematicas´ (CSIC-UAM-UC3M-UCM) and Departa- mento de Matematicas,´ Universidad Autonoma´ de Madrid, 28049 Madrid, Spain E-mail address: [email protected]

A. Iosevich: Department of Mathematics, University of Rochester, Rochester, NY E-mail address: [email protected]

B. Lund: Department of Computer Science, Rutgers, The State University of New Jer- sey, NJ E-mail address: [email protected]

O. Roche-Newton: Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, 4040 Linz, Austria E-mail address: [email protected]

M. Rudnev: School of Mathematics, University Walk, Bristol, BS8 1TW E-mail address: [email protected]