An improved incidence bound for fields of prime order

European Journal of Combinatorics 52 (2016) 136–145

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An improved incidence bound for fields of prime order Timothy G.F. Jones School of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom

article

info

Article history: Received 16 November 2011 Accepted 17 September 2015 Available online 1 November 2015

abstract Let P be a set of points and L a set of lines in F2p , with |P |, |L| ≤ N and 3

1

N < p. We show that P and L generate no more than CN 2 − 806 +o(1) incidences for some absolute constant C . This improves on the 3

−

1

previously best-known bound of CN 2 10,678 . © 2015 Published by Elsevier Ltd.

1. Introduction Throughout this paper we use X = Ω (Y ), Y = O(X ), and Y ≪ X all to mean that there is an absolute constant C with Y ≤ CX . We shall write X ≈ Y if X ≪ Y and Y ≪ X . 1.1. Incidences This paper is about counting incidences between points and lines in a plane. A point is incident to a line if it lies on that line. Incidences are counted with multiplicity, in the sense that several lines incident to the same point determine several incidences, and vice versa. We are interested in knowing the maximum number of incidences between a set P of points and a set L of lines, say with |P |, |L| ≤ N. Certainly this cannot exceed N 2 . But using the Cauchy–Schwarz inequality and the fact that two distinct points determine a line, it is straightforward to see that it is in fact O(N 3/2 ). So, writing I (P , L) for the number of incidences between P and L, non-trivial incidence bounds are  of the form I (P , L) = O N 3/2−ϵ with ϵ > 0. The larger the value of ϵ , the stronger the bound.

E-mail address: [email protected]. http://dx.doi.org/10.1016/j.ejc.2015.09.004 0195-6698/© 2015 Published by Elsevier Ltd.

T.G.F. Jones / European Journal of Combinatorics 52 (2016) 136–145

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A natural example shows that the strongest result that can be hoped for is ϵ = 1/6. Progress towards achieving this depends on the ambient field over which points and lines are defined. In the case of the plane R2 , the best possible result was obtained by Szemerédi and Trotter [13]: Theorem 1 (Szemerédi–Trotter). Let P be a set of points and L a set of lines in R2 with |P |, |L| ≤ N. Then I (P , L) = O(N 4/3 ). This result was generalised to C2 by Tóth [15], and a near-sharp generalisation to higher dimensional points and varieties was recently given by Solymosi and Tao [12]. When working over finite fields, some additional condition must be imposed if we are to prove nontrivial incidence bounds. Otherwise we would be free to take P to be the entire plane, where only trivial bounds would be possible. Vinh [16] has given a nontrivial incidence bound for sets that are a large part, but not all of, of the plane. Theorem 2 (Vinh). Let Fq be the finite field of order q. Let P be a set of points and L a set of lines in F2q 3

γ

with |P |, |L| ≤ N and p1+γ < N < p2−γ for 0 < γ < 1. Then I (P , L) = O(N 2 − 4 ). In the case of smaller sets, Helfgott and Rudnev [4] obtained the following result for the finite field of prime order p. Theorem 3 (Helfgott–Rudnev). Let Fp be the finite field of prime order p. Let P be a set of points and L a 3

set of lines in F2p with |P |, |L| ≤ N and N < p. Then I (P , L) = O(N 2

− 10,1678

).

This followed work of Bourgain, Katz and Tao [2], which established the existence of a non-zero ϵ so long as N < p2−δ(ϵ) , but did not quantify it. The present author [5] extended Theorem 3 to a finite field Fq of general order, subject to analogous conditions to prevent P from being a large part of a subplane, and with a slightly weaker exponent. This paper proves the following theorem, which improves on Theorem 3: Theorem 4. Let Fp be the finite field of prime order p. Let P be a set of points and L a set of lines in F2p with 3

1

|P |, |L| ≤ N and N < p. Then I (P , L) = O(N 2 − 806 +o(1) ). As will be seen, the proof of Theorem 4 uses finite field sum–product estimates as a ‘black box’. It is likely that further improvements can be obtained by unpacking the sum–product proof and exploiting its use of multiplicative energy and covering arguments to bypass some costly Balog–Szemerédi–Gowers type refinements. It should also be remarked that by combining recent sum–product work of Li and Roche-Newton [7] with the approach in [5], a result comparable to Theorem 4 should hold for a general finite field Fq , subject to appropriate non-degeneracy conditions. 1.2. Relation to sum–product estimates If A and B are subsets of a field F , then we will write A + B = {a + b : a ∈ A, b ∈ B} . A · B = {ab : a ∈ A, b ∈ B} . These definitions extend analogously to subtraction and division. Erdős and Szemerédi conjectured that any finite set A ⊆ R must satisfy max {|A + A|, |A · A|} ≫ϵ |A|2−ϵ for any ϵ > 0. Nontrivial lower bounds on max {|A + A|, |A · A|} are generally called sum–product estimates. The strongest-known result is due to Solymosi [11], who obtained 4

max {|A + A|, |A · A|} ≫ |A| 3 −o(1) .

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This result has been extended to C by Konyagin and Rudnev [6]. There are also interesting variations on this theme, for example in work of Li and Roche-Newton [8] who showed that if f is a strictly convex function then 24

|A + f (A)| ≫ |A| 19 −o(1) for any finite set A of reals. Incidence and sum–product estimates are closely linked. For example the Szemerédi–Trotter theorem was used to establish the previously best-known sum–product result over R of max{|A + A|, |A · A|} ≫ |A|14/11−o(1) . This was again due to Solymosi [10] and built on previous work of Elekes [3]. Similarly, the sum–product results of Konyagin and Rudnev in C depend on incidence results in C2 , and the convexity result of Li and Roche-Newton depends on incidences in R2 . In the other direction, the Helfgott–Rudnev and Bourgain–Katz–Tao incidence bounds over Fp were established using finite field sum–product results and methods. True to this tradition, the proof of Theorem 4 relies on sum–product estimates over Fp . The strongest known sum–product estimate over Fp , due to Rudnev [9], is 12

max {|A + A|, |A · A|} ≫ |A| 11 −o(1)

√

so long as |A| < p. As Rudnev’s paper describes, the result can be expressed as a ‘difference–ratio’ estimate as well, in which case the o(1) in the exponent disappears. We will use the following formulation: Theorem 5 (Rudnev). Let A ⊆ Fp with |A| <

√

p. Then max {|A − A|, |A/A|} ≫ |A|12/11 .

2. Standard lemmata This section records some results that will be used in the proof of Theorem 4. 2.1. Pigeonholing results The first two lemmata are standard pigeonholing techniques. I (P ,L) 2|P |

Lemma 6. Let P1 be the set of points in P incident to at least

lines in L. Then

I (P1 , L) ≈ I (P , L). Similarly, if L1 is the set of lines in L incident to at least

I (P ,L) 2|L|

points in P then

I (P , L1 ) ≈ I (P , L). Proof. We prove the result for points; the version for lines is similar. Let P2 be the set of points in P I (P ,L) incident to at most 2|P | lines in L. Then I (P2 , L) =



# {l ∈ L incident to p} ≤ |P2 |

p∈P2

I (P , L) 2|P |

≤

I (P , L) 2

.

Since I (P , L) = I (P1 , L) + I (P2 , L) it follows that I (P1 , L) ≥ as required.

I (P , L) 2







Lemma 7. Let P and L be a set of points and lines respectively. For each p ∈ P let Pp = q ∈ P : lpq ∈ L , where lpq is the line determined by the points p and q. Let P1 be a subset of P with the property that |Pp | ≥ α

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for any p ∈ P1 . Then at least one of the following is true: |P |2

1. α 2 ≪ |P | . 1 2 2. There exist distinct p1 , p2 ∈ P1 such that |Pp1 ∩ Pp2 | ≫ |αP | . Proof. By hypothesis and Cauchy–Schwarz,

1/2

 |P1 |α ≤



1/2



|Pp | ≤ |P |

|Pp1 ∩ Pp2 |

.

p1 ,p2 ∈P1

p∈P1

We therefore have



|P1 |2 α 2 ≤ |P |2 |P1 | + |P |

|Pp1 ∩ Pp2 |.

p1 ̸=p2 ∈P1

It is then either the case that

α2 ≪

|P |2 , |P1 |

in which case the first conclusion holds, or that

|P1 |2 α 2 ≪ |P |



|Pp1 ∩ Pp2 |,

p1 ̸=p2 ∈P1

in which case there exist p1 ̸= p2 such that the second holds.



2.2. Balog–Szemerédi–Gowers We will be interested in partial difference and ratio sets E

A − B = {a − b : (a, b) ∈ E } E

A / B = {a/b : (a, b) ∈ E } where E ⊆ A × B. We will wish to apply sum–product (or strictly speaking, difference–ratio) estimates to some partial difference and ratio sets. However to do this we need to work with complete difference and ratio sets. The traditional machinery for making such a transition is the Balog–Szemerédi–Gowers theorem. We employ a variation, itself a straightforward modification of one due to Bourgain and Garaev [1]: Lemma 8. Let A, B ⊆ Fp and E ⊆ A × B. Then there exists A′ ⊆ A with |E |

1. |A′ | ≫ |B| |A|4 |B|3 | A′ −A′ | | A′ / A′ | 2.  4 ,  4 ≪ |E |5 . E  E    A−B

A/B  

2.2.1. Proof of Lemma 8 The proof of Lemma 8 follows the approach in [1] closely, making only a slight technical modification. The following lemma can be found as Lemma 6.19 in the book of Tao and Vu [14]. Lemma 9. Let E ⊆ A × B. For any a ∈ A, write N (a) for the set of b ∈ B with (a, b) ∈ E. Then for any ϵ > 0 there exists A′ ⊆ A with |A′ | ≫ ||EB|| such that, for at least (1 − ϵ)|A′ |2 of the pairs (a′1 , a′2 ) ∈ A′ × A′ , we have 2   N (a′ ) ∩ N (a′ ) ≥ ϵ|E | . 1 2 2 2|A| |B|

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We apply it to obtain: Lemma 10. Let G(A, B, E ) be a bipartite graph. Then there exists A′ ⊆ A with |A′ | ≫ |B| such that every   |E |

pair of elements from A′ × A′ is connected by Ω

|E |5

|A|4 |B|3

paths of length four in E.

Proof. Say that (a1 , a2 ) ∈ A × A is good if

|N (a1 ) ∩ N (a2 )| ≥ 0.05

|E |2 . |A|2 |B|

By the previous lemma we can find |A′ | ≫ |B| such that 0.9|A′ |2 of pairs from A′ are good. Given a′1 ∈ A′ denote by Ia′ the set of elements a′2 ∈ A′ for which (a′1 , a′2 ) is good. Then we have |E |

1



|Ia′ | ≥ 0.9|A | . ′ 2

a′ ∈A′

So by popularity pigeonholing there exists A′′ ⊆ A′ with |A′′ | ≫ |A′ | such that |Ia | ≥ 0.7|A′ | for every a ∈ A′′ . So for any pair a1 , a2 ∈ A′′ we have |Ia1 ∩ Ia2 | ≫ |A′ | and so there are Ω (|A′ |) elements c ∈ A′ for which

|N (a1 ) ∩ N (c )|, |N (a2 ) ∩ N (c )| ≫ This means that there are Ω



|E |5 |A|4 |B|3



|E |2 . |A|2 |B|

paths of length four in E connecting a1 and a2 , as required.



We now prove Lemma 8. Let A′ be as provided in Lemma 10. For each α, β ∈ A′ we have, # {(a, b1 , b2 ) ∈ A × B × B : (α, b1 ), (a, b1 ), (a, b2 ), (β, b2 ) ∈ E } ≫

|E |5 . |A|4 |B|3

Now it is clear that

α − β = (α − b1 ) − (a − b1 ) + (a − b2 ) − (β − b2 ) α b1 a b2 α = · · · . β b1 a b2 β So for all α, β ∈ A′ we have



E

# (s, t , u, v) ∈ (A − B) : s − t + u − v = α − β



4

E

# (s, t , u, v) ∈ (A \ B)4 :

st uv

=

α β

 ≫

 ≫

|E |5 |A|4 |B|3

|E |5 . |A|4 |B|3

Summing over all elements of A − A′ and A′ /A′ respectively, we obtain ′

   E 4 |E |5 ′ ′  |A − A | ≪ A − B |A|4 |B|3    E 4 |E |5 ′ ′ A / B . | A / A | ≪   |A|4 |B|3 Rearranging gives the statement of Lemma 8. 3. Proof of Theorem 4 We now begin the proof 4. Without loss of generality we may assumethat every point  of Theorem  

in P is incident to only O

N2 I (P ,L)

lines in L and every line in L is incident to only O

N2 I (P ,L)

points in

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141

2

P. Indeed, if P+ is the set of points incident to at least I (3N lines then, writing δpl = 1 if p ∈ l and 0 P ,L ) otherwise, we have I (P+ , L) =



δpl ≤

I ( P , L)  

p∈P+ l∈L

3N 2

p∈P+ l1 ,l2 ∈L

δpl1 δpl2 ≤

2I (P , L) 3

and so P+ can be discarded without affecting incidences by more than a multiplicative constant. The argument for lines is similar.   We may also apply Lemma 6 to assume that every point in P is incident to at least Ω

I (P ,L) N

lines

in L. 3.1. Obtaining partial difference and ratio sets Our first aim is to show that if we have too many incidences then there must exist large A, B ⊆ Fp E

E

and a large E ⊆ A × B such that A − B and A / B are both small. We will then be able to apply Lemma 8 to transition from partial to complete difference and ratio sets, and then apply Theorem 5 to obtain a bound on the number of incidences. We shall accomplish this by making successive refinements to P, to construct a point set of a particular form, and then applying a projective transformation. 3.1.1. First refinement By a dyadic pigeonholing we may find a subset P1 ⊆ P and an integer K such that every point in P1 is incident to between K and 2K lines in L, and I (P , L)

I (P1 , L) ≈ |P1 |K ≈

log N

.

In view of our earlier assumptions about the number of lines incident to points in P, we know that I (P , L) N

≪K ≪

N2 I ( P , L)

.

By Lemma 6 there is a set L1 of lines in L that are each incident to

Ω



I (P1 , L)



|L|

=Ω



I ( P , L)



N log N

points in P1 such that I (P1 , L1 ) ≈ I (P1 , L) ≈ |P1 |K . Applying Lemma 6 again, there is a set P2 of points in P1 that are each incident to

Ω



I (P1 , L1 )



|P1 |

=Ω



|P1 |K |P1 |



= Ω (K )

lines in L1 such that I (P2 , L1 ) ≈ I (P1 , L1 ) ≈ |P1 |K . We have

|P2 | ≫

I (P , L)2 N 2 log N

since I (P , L) log N

≪ I (P1 , L) ≪ I (P2 , L1 ) ≪

|P2 |N 2 . I ( P , L)

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T.G.F. Jones / European Journal of Combinatorics 52 (2016) 136–145

3.1.2. Selecting two special points For each p ∈ P2 let Pp = q ∈ P1 : lpq ∈ L1 .





I (P ,L)

Since p is incident to Ω (K ) lines in L1 and each line is incident to at least Ω ( N log N ) points in P1 we have KI (P , L) |Pp | ≫ . N log N Applying Lemma 7 and the lower bound on |P2 | we see that either K 2 I (P , L)2

N 4 log N |P1 |2 ≪ |P2 | I (P , L)2 log N or that there exist distinct p1 , p2 ∈ P2 such that 2 2 2 2   Pp ∩ Pp  ≫ K I (P , L) ≥ K I (P , L) . 1 2 |P2 |N 2 log2 N N 3 log2 N 2

N2

≪

In the former case we obtain I (P , L) ≪ N 4/3+o(1) and are done already. So we proceed on the basis of the latter. 3.1.3. Another refinement Pick any Q ⊆ Pp1 ∩ Pp2 with

|Q | ≈

K 2 I (P , L)2 N 3 log2 N

.

Since Q ⊆ P1 we have I (Q , L) ≈ K |Q | ≈

K 3 I (P , L)2 N 3 log2 N

.

Moreover, since Q ⊆ Pp1 ∩ Pp2 , it is supported over only O(K ) lines through each of p1 and p2 . Apply Lemma 6 to find a set J ⊆ L of lines that are each incident to

Ω



I ( Q , L)



|L|

=Ω



K 3 I (P , L)2



N 4 log2 N

points in Q with I (Q , J ) ≈ I (Q , L) ≈

K 3 I (P , L)2 N 3 log2 N

.

Apply Lemma 6 again to find a set Q1 ⊆ Q of points that are each incident to

Ω



I (Q , J )

|Q |



=Ω



|Q |K |Q |



= Ω (K )

lines in J with I (Q1 , J ) ≈ I (Q , J ) ≈

K 3 I (P , L)2 N 3 log2 N

We have

|Q1 | ≫

K 3 I (P , L)3 N 5 log2 N

since I (Q , L) ≪ I (Q1 , J ) ≪

|Q1 |N 2 . I ( P , L)

.

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3.1.4. Picking a special line incident to p1 but not p2 Let Jp1 be the set of lines in J that are incident to p1 . Because J ⊆ L and p1 ∈ P1 , we know that |Jp1 | ≪ K and therefore by the lower bound on |Q1 | that K 3 I (P , L)3 N 5 log2 N

≪ |Q1 | ≪



|Q1 ∩ l|.

l∈Jp1

Each line in J is incident to at most O



N2 I (P ,L)



points in P and therefore in Q1 . This quantity is dominated

by the left hand side of the estimate above. Indeed, if it were not then by the lower bound on K we would have I (P , L) ≪ N 10/7−o(1) and be done already. So we may discard a single line from Jp1 without affecting the estimate. We can therefore pick a line l in Jp1 that is not incident to p2 such that

|Q1 ∩ l| ≫

K 2 I (P , L)3 N 5 log2 N

.

We take Q2 to be the set of points in Q1 incident to l, i.e. Q2 = Q1 ∩ l. 3.1.5. Picking two further points on the special line For each p ∈ Q2 let Qp = q ∈ Q : lpq ∈ J .





Since  3p ∈ Q2 2 ⊆ Q1 we know that it is incident to Ω (K ) lines in J. Because each line in J is incident to Ω K 4I (P ,2L) points in Q1 we therefore have N log N

|Qp | ≫

K 4 I (P , L)2 N 4 log2 N

.

Applying Lemma 7 again, and the lower bound on |Q2 | = |Q1 ∩ l| we see that either I (P , L)12 N 16 log4 N

≪

K 8 I (P , L)4 N 8 log4 N

≪

|Q |2 N 9 log2 N ≪ |Q2 | I (P , L)5

or that there are distinct p3 , p4 ∈ Q2 , not equal to p1 , p2 , such that

|Qp3 ∩ Qp4 | ≫

K 8 I (P , L)4

|Q |

N8

4

log N

≫

K 6 I (P , L) N 5 log2 N

.

In the former case, we have I (P , L) ≪ N 25/17−o(1) and are done. So we proceed on assumption of the latter. Define R = Qp3 ∩ Qp4 so that

|R| ≫

K 6 I (P , L)2 N 5 log2 N

and R is supported on O(K ) lines through each of p1 , p2 , p3 , p4 . 3.1.6. Projective transformation Since p3 , p4 ∈ Q2 they are collinear with p1 but not p2 . Pick a projective transformation τ that sends their common line to the line at infinity, in such a way that lines through p3 and p4 are parallel to the horizontal and vertical co-ordinate axes respectively. If we define E = τ (R) then this means that E ⊆ A × B for A, B ⊆ Fp . By appending points if necessary we may assume

|A|, |B| ≈ K .

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By discarding points we may assume that

|E | ≈

K 6 I (P , L)2 N 5 log2 N

.

Since p1 is collinear with p3 and p4 , lines through τ (p1 ) are all parallel, but not parallel to the vertical or horizontal axes. So there exists z ̸∈ {0, ∞} such that E is supported on only O(K ) lines of gradient z. Since p2 is not collinear with p3 , the map τ sends it to the affine plane, and so E is supported on O(K ) lines all passing through τ (p2 ). Note that these properties are invariant under translation of the plane and scaling of a single coordinate axis in the following sense. First, if T is some translation of the plane then T (E ) is supported on O(K ) lines through T (τ (p2 )), and still covered by O(K ) lines of gradient z. So we may assume that τ (p2 ) is the origin. Second, if δλ is the transformation that sends (x, y) to (x, λy), then (x, y) lies on the line y = mx + c if and only if δλ (x, y) lies on the line y = (λm)x + λc. So δλ (E ) is supported on O(K ) lines of gradient λz and O(K ) lines through the origin. By taking λ = 1/z we may assume therefore that z = 1. So without loss of generality we may assume that E is supported on O(K ) lines through the origin and O(K ) lines of gradient 1. In other words

     E   E  A / B , A − B ≪ K .     3.2. Concluding the proof To apply Lemma 8 we note that I (P , L)7 K 5 I (P , L)2 |E | ≈ 5 2 ≫ 10 2 |B| N log N N log N and

     E 4  E 4 4 3   |A| |B| A − B |A| |B| A / B N 25 log10 N N 44 log10 N , ≪ ≪ . |E |5 |E |5 K 19 I (P , L)19 4

3

So there exists A′ ⊆ A with

|A′ | ≫

I (P , L)7 N 10 log2 N

and

|A′ − A′ |, |A′ /A′ | ≪

N 44 log10 N I (P , L)19

.

We are free to pass to a subset of A′ if we wish, and so assume that

|A′ | ≈

I (P , L)7 N 10 log2 N

.

By the trivial incidence bound I (P , L) ≪ N 3/2 we have I (P , L)7 2

N 10 log N

≪

N 1/2 log2 N

< p1/2

and so may apply Theorem 5 to A′ since |A′ | < p1/2 . This yields 604

3

1

I (P , L) ≪ N 303 +o(1) = N 2 − 806 +o(1) , which completes the proof of Theorem 4.

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