Some multiplicative equations in finite fields
Introduction
For a prime number q and integer n consider the finite field with elements. For a subset we define the multiplicative energy of to count the number of solutions to the equation In this paper we consider estimating for certain sets 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 states that for any ε there exists some such that if then with the condition that if then does not have a large intersection with any proper subfield, where and denote the sum and product set 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 one may take any fixed . We expect this conjecture to remain true over finite fields with suitable size restrictions on and the intersection of 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 , prime fields and general finite fields.
A typical approach to the sum-product problem is to estimate the multiplicative energy of a set in terms of the size of the sumset since it follows from the Cauchy-Schwarz inequality For sets satisfying we expect that from which it would follow that This is known to hold over 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 form and say that is proper if By Frieman's theorem, see for example [25, Chapter 5], every set 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 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 be a basis for as a vector space over and define the box The first estimates for 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 . Davenport and Lewis [10] provided the first estimate for uniform with respect to the basis 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 bound in the special case that Recently Gabdullin [13] has extended Konyagin's estimate to arbitrary boxes when . In this paper we show Konyagin's estimate holds with the weaker condition 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 is a basis for as a vector space over . For two n-tuples of positive integers and we let B denote the box If satisfy and then we have Corollary 2 Let q be prime, n a positive integer and suppose
Background from the geometry of numbers
The following is Minkowski's second theorem, for a proof see [25, Theorem 3.30]. Lemma 6 Suppose is a lattice, a convex body and let denote the successive minima of Γ with respect to D. Then we have Lemma 7 Suppose is a lattice, a convex body and let denote the successive minima of Γ with respect to D. Then we have
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 . There exists a nontrivial integral solution to the system of equations satisfying where A denotes the matrix with -th entry and denotes the transpose of A.
Lemma 10 Let q be prime, n an integer and integers satisfying and Suppose is a basis for as a vector
Proof of Theorem 1
For we let count the number of solutions to the equation so that We define the lattice and the convex body For any two points and satisfying (22) we have and hence Let so that Let
Multiplicative energy of generalized arithmetic progressions
For two d-tuples of real numbers and we define the Bohr set and for a generalized arithmetic progression given by we let count the number of solutions to the congruence
The following is based on some ideas of Ayyad, Cochrane and Zheng [1]. Lemma 12 With notation as above, suppose that is proper. Then we have Proof Let denote the
Proof of Theorem 3
We first note the assumption is proper implies that and in particular Hence by Lemma 12 it is sufficient to show that Suppose is such that the expression occurring in (32) is maximum for some . We have where counts the number of solutions to the congruence We define and for each let denote
Proof of Corollary 4
Let count the number of solutions to with , so that Let denote the progression and suppose counts the number of solutions to the equation If then and hence and the result follows since is the union of at most proper progressions of the form covered by Theorem 3.
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