Generalized incidence theorems, homogeneous forms and sum–product estimates in finite fields

Abstract

In recent years, sum-product estimates in Euclidean space and finite fields have been studied using a variety of combinatorial, number theoretic and analytic methods. Erdos type problems involving the distribution of distances, areas and volumes have also received much attention. In this paper we prove a relatively straightforward function version of an incidence results for points and planes previously established in [10] and [12]. As a consequence of our methods, we obtain sharp or near sharp results on the distribution of volumes determined by subsets of vector spaces over finite fields and the associated arithmetic expressions.In particular, our machinery enables us to prove that if E ⊂ F d q , d ≥ 4, the ddimensional vector space over a finite field F q , of size much greater than q d 2 , and if E is a product set, then the set of volumes of d-dimensional parallelepipeds determined by E covers F q . This result is sharp as can be seen by taking E to equal to A × A × · · · × A, where A is a sub-field of F q of size √ q. In three dimensions we establish the same result if |E| q 15 8 . We prove in three dimensions that the set of volumes covers a positive proportion of F q if |E| ≥ Cq 3 2 . Finally we show that in three dimensions the set of volumes covers a positive proportion of F q if |E| ≥ Cq 2 , without any further assumptions on E, which is again sharp as taking E to be a 2-plane through the origin shows.

On an Application of Guth–Katz Theorem

“…For finite field versions of these problems see, for example, [12] and [2]. In all of these instances, the exponents are not optimal.…”

Additive Combinatorics over Finite Fields: New Results and Applications

“…Furthermore, Schoen & Shkredov [167] have successfully used a "cubic" generalization of the energy. We also have had to leave out such exciting areas of additive combinatorics in finite fields as • the Erdős distance problem [83,94,117,130,132,144,145] as well as its modification in some other settings (distinct volumes, configurations, and so on defined by arbitrary sets in F n q ) and metrics [14,64,142,195,200,202,203,205]; • the Kakeya problem and other related problems about the directions defined by arbitrary sets in vector spaces over a finite field, see [84-86, 88, 128, 131, 151]; • estimating the size of the sets in a finite field that avoid arithmetic or geometric progressions, sum sets and similar linear and non-linear relations; in particular these results include finite field analogues of the Roth and Szemerédi theorems, see [1,6,8,12,77,81,112,113,153,154,158,181]; • estimating the size of the sets in vector spaces over a finite field that define only some restrictive geometric configurations such as integral distances, acute angles, and pairwise orthogonal systems, see [75,133,134,183,194,204]; • distribution of the values of determinants and permanents of matrices with entries from general sets, see [74,196,197]; and several others.…”

Sets with integral distances in finite fields

“…The first term on the right hand side is q n−1 δ(v) by using (7). Denoting by q −1 T 2 the second term, one has…”

“…. , x n ∈ E. See [7] for a recent study of the volume sets. Now, given t ∈ F q we define the undirected graph G t as a graph whose vertices are labelled by vectors x ∈ F n q and the vertices x, y are connected if and only if x − y ∈ S n (t).…”