Some multiplicative equations in finite fields

https://doi.org/10.1016/j.ffa.2021.101883Get rights and content

Abstract

In this paper we consider estimating the number of solutions to multiplicative equations in finite fields when the variables run through certain sets with high additive structure. In particular, we consider estimating the multiplicative energy of generalized arithmetic progressions in prime fields and of boxes in arbitrary finite fields. We obtain sharp bounds in more general scenarios than previously known. Our arguments extend some ideas of Konyagin and Bourgain and Chang into new settings.

Introduction

For a prime number q and integer n consider the finite field Fqn with qn elements. For a subset AFqn we define the multiplicative energy E(A) of A to count the number of solutions to the equationa1a2=a3a4,a1,a2,a3,a4A. In this paper we consider estimating E(A) for certain sets A with large additive structure. In particular, we consider the case of boxes in arbitrary finite fields and generalized arithmetic progressions in prime fields. These two problems may be considered as extreme cases of the sum-product phenomenon of Erdös and Szemerédi [12], established in the setting of prime fields by Bourgain, Katz and Tao [6] and arbitrary finite fields by Katz and Shen [16]. The sum-product theorem over Fqn states that for any ε there exists some δ>0 such that if |A|q(1ε)n thenmax{|AA|,|A+A|}|A|1+δ, with the condition that if n2 then A does not have a large intersection with any proper subfield, where AA and A+A denote the sum and product setAA={a1a2:a1,a2A},A+A={a1+a2:a1,a2A}. An important factor in this problem is how large one may take δ in (1). Erdös and Szemerédi [12] conjectured that for any set of integers A one may take any fixed δ<1. We expect this conjecture to remain true over finite fields with suitable size restrictions on A and the intersection of A with proper subfields. Current techniques are still far from resolving this conjecture and we refer the reader to [22], [23] and [17], [21] for the current best quantitative results for sum product over R, prime fields and general finite fields.

A typical approach to the sum-product problem is to estimate the multiplicative energy of a set A in terms of the size of the sumset A+A since it follows from the Cauchy-Schwarz inequality|AA||A|4E(A). For sets A satisfying|A+A||A|, we expect thatE(A)|A|2+o(1), from which it would follow that|AA||A|2ε. This is known to hold over R by a result of Elekes and Ruzsa [11], see also [8], although still open in the case of finite fields and we refer the reader to [20] for the sharpest results in the setting of small sumset in prime fields.

In this paper we consider the problem of obtaining estimates of the strength (3) under the condition (2) in the setting of finite fields and obtain some new instances of when this bound holds. An important class of sets with small sumset are generalized arithmetic progressions, which are defined as sets of the formA={α1h1++αdhd+β:1hiHi}, and say that A is proper if|A|=H1Hd. By Frieman's theorem, see for example [25, Chapter 5], every set A satisfying (2) is dense in some proper generalized arithmetic progression and a natural approach to extending the result of Elekes and Ruzsa [11] into finite fields is to show that (3) holds for proper generalized arithmetic progressions. We take a step forward in this direction and give the expected upper bound for E(A) for a certain family of generalized arithmetic progressions, see Theorem 3 below. Roughly speaking, our result holds for generalized arithmetic progressions which are smaller portions of proper generalized arithmetic progressions.

We also consider estimating the multiplicative energy of boxes in arbitrary finite fields. Let ω1,,ωn be a basis for Fqn as a vector space over Fq and define the boxB={ω1h1++ωnhn:Mi<hiMi+Hi}. The first estimates for E(B) were motivated by the problem of extending the Burgess bound into arbitrary finite fields and are due to Burgess [7] and Karatsuba [14], [15] although the results of Burgess and Karatsuba are not uniform with respect to the basis ω1,,ωn. Davenport and Lewis [10] provided the first estimate for E(B) uniform with respect to the basis ω1,,ωn although their bound is quantitatively much weaker than that of Burgess and Karatsuba. The estimate of Davenport and Lewis was improved by Chang [9] using techniques from Additive Combinatorics which was further improved by Konyagin [18] who showed the expected upper boundE(B)|B|2+o(1), in the special case thatH1=H2==Hn. Recently Gabdullin [13] has extended Konyagin's estimate to arbitrary boxes when n=2,3. In this paper we show Konyagin's estimate holds with the weaker conditionmaxHiq1/nminHi, for arbitrary n. We follow Konyagin's strategy which is based on considering the successive minima of a certain family of lattice and their duals and our main novelty for this section comes from establishing certain inequalities for these successive minima by using Siegel's lemma.

Finally we draw some comparisons between our argument for generalized arithmetic progressions and Konyagin's approach [18], further developed by Bourgain and Chang [5] to deal with multiplicative equations with systems of linear forms. Both Konyagin and Bourgain and Chang reduce the problem to a lattice point counting problem over a family of lattices. An important feature of these families is that they are self dual which allows control of the successive minima via transference theorems. In order to reduce the problem of multiplicative energy of generalized arithmetic progressions into a lattice point counting problem with the same symmetry as in [5], [18] we first expand into additive characters and considering the sets of large Fourier coefficients, this allows a reduction of the problem into multiplicative equations with generalized arithmetic progressions and Bohr sets and this form of the problem has suitable symmetry.

Section snippets

Main results

Theorem 1

Let q be prime, n a positive integer and suppose ω1,,ωn is a basis for Fqn as a vector space over Fq. For two n-tuples of positive integers H=(H1,,Hn) and M=(M1,,Mn) we let B denote the boxB={ω1h1++ωnhn:Mi<hiMi+Hi}. If H1,,Hn satisfyHnHn1H1q,k=1i1HkqHiiHi1,2in, andHni+1iqHnk=ni+2nHk,2in, then we haveE(B)|B|4qn+|B|2(log|B|)n.

We may put the conditions on H1,,Hn occurring in Theorem 1 in the following simpler form.

Corollary 2

Let q be prime, n a positive integer and suppose ω1,,ωn

Background from the geometry of numbers

The following is Minkowski's second theorem, for a proof see [25, Theorem 3.30].

Lemma 6

Suppose ΓRd is a lattice, DRd a convex body and let λ1,,λd denote the successive minima of Γ with respect to D. Then we haveVol(D)Vol(Rd/Γ)1λ1λdVol(D)Vol(Rd/Γ).

For a proof of the following, see [3, Proposition 2.1].

Lemma 7

Suppose ΓRd is a lattice, DRd a convex body and let λ1,,λd denote the successive minima of Γ with respect to D. Then we have|ΓD|j=1dmax(1,1λj).

For a lattice Γ and a convex body D we

Multiplicative energy of boxes in finite fields

The following version of Siegel's Lemma is due to Bombieri and Vaaler [4].

Lemma 9

Let M and L be integers with M>L. There exists a nontrivial integral solution (t1,,tM) to the system of equationsa,1t1++a,MtM=0=1,,L, satisfyingmax1mM|tm||det(AAt)|1/2(ML), where A denotes the matrix with (,m)-th entry a,m and At denotes the transpose of A.

Lemma 10

Let q be prime, n an integer and H1,,Hn integers satisfyingHnHn1H1q, andk=1i1HkqHiiHi1,2in. Suppose ω1,,ωn is a basis for Fqn as a vector

Proof of Theorem 1

For zFqn we let I(z) count the number of solutions to the equationz(ω1x1++ωnxn)=ω1y1++ωnyn,Mixi,yiMi+Hi, so thatE(B)=zFqnI(z)2zFqnz0I(z)2+|B|2. We define the latticeΓ(z)={(x1,,xn,y1,,yn)Z2n:z(ω1x1++ωnxn)=ω1y1++ωnyn}, and the convex bodyD={(x1,,xn,y1,,yn)R2n:|xi|,|yi|Hi1in}. For any two points (x1,,yn) and (x1,,yn) satisfying (22) we have(x1x1,,ynyn)Γ(z)D, and henceE(B)zFqnz0|Γ(z)D|2+|B|2. LetΩ={zFqn/{0}:Γ(z)D{0}}, so thatE(B)zΩ|Γ(z)D|2+|B|2. Let λ1(

Multiplicative energy of generalized arithmetic progressions

For two d-tuples of real numbers ε=(ε1,,εd) and α=(α1,,αd) we define the Bohr setB(α,ε)={1xq1:αixqεii=1,,d}, and for a generalized arithmetic progression A given byA={α1h1++αdhd:1hiH}, we let E(A,ε) count the number of solutions to the congruencea1b1a2b2modqa1,a2A,b1,b2B(α,ε).

The following is based on some ideas of Ayyad, Cochrane and Zheng [1].

Lemma 12

With notation as above, suppose that A is proper. Then we haveE(A)=|A|4q+O((logH)2dqmax1/Hεi1E(A,ε)(ε1εd)2).

Proof

Let A(x) denote the

Proof of Theorem 3

We first note the assumptionA={α1h1++αdhd:|hi|H2}, is proper implies thatHq1/2d, and in particularH4dqH2d. Hence by Lemma 12 it is sufficient to show thatmax1/Hεi1E(α,ε)(ε1εd)2qH2d(logH). Supposeε=(δ1H,,δdH), is such that the expression occurring in (32) is maximum for some δ1,,δd1. We haveE(A,ε)=ω=1q1I(ω)2, where I(ω) counts the number of solutions to the congruenceaωbmodqaA,bB(α,ε). We defineL={(y1,,yd)Zd:1xqsuch thatyiαixmodq}, and for each 1ωq1 let Γ(ω) denote

Proof of Corollary 4

Let I(λ) count the number of solutions toa1λa2modq, with a1,a2A, so thatE(A)=λI(λ)2. Let A0 denote the progressionA0={α1h1++αdhd:|hi|H}, and suppose I0(λ) counts the number of solutions to the equationa1λa2,a1,a2A0. If I(λ)0 thenI(λ)I0(λ), and henceE(A)λI0(λ)2+|A|2E(A0)+|A|2, and the result follows since A0 is the union of at most 2d proper progressions of the form covered by Theorem 3.

References (25)

  • A. Ayyad et al.

    The congruence x1x2x3x4modp, the equation x1x2=x3x4 and mean values of character sums

    J. Number Theory

    (1996)
  • J. Bourgain et al.

    On a multilinear character sum of Burgess

    C. R. Acad. Sci. Paris, Ser. I

    (2010)
  • O. Roche-Newton et al.

    New sum-product type estimates over finite fields

    Adv. Math.

    (2016)
  • W. Banaszczyk

    Inequalities for convex bodies and polar reciprocal lattices in Rn

    Discrete Comput. Geom.

    (1995)
  • U. Betke et al.

    Successive-minima-type inequalities

    Discrete Comput. Geom.

    (1993)
  • E. Bombieri et al.

    On Siegel's lemma

    Invent. Math.

    (1983)
  • J. Bourgain et al.

    A sum-product estimate in finite fields and their applications

    Geom. Funct. Anal.

    (2004)
  • D.A. Burgess

    Character sums and primitive roots in finite fields

    Proc. Lond. Math. Soc. (3)

    (1967)
  • M.C. Chang

    Factorization in generalized arithmetic progressions and applications to the Erdös-Szemerédi sum-product problems

    Geom. Funct. Anal.

    (2003)
  • M.C. Chang

    On a question of Davenport and Lewis and new character sum bounds in finite fields

    Duke Math. J.

    (2008)
  • H. Davenport et al.

    Character sums and primitive roots in finite fields

    Rend. Circ. Mat. Palermo (2)

    (1963)
  • G. Elekes et al.

    Few sums, many products

    Studia Sci. Math. Hung.

    (2003)
  • View full text