Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdős-Falconer distance conjecture

Abstract

Abstract. We prove a pointwise and average bound for the number of incidences between points and hyperplanes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering F q , the finite field with q elements, by A · A + · · · + A · A, where A is a subset F q of sufficiently large size. We also use the incidence machinery and develop arithmetic constructions to study the Erdős-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdős-Falconer distance conjecture does not hold in this setting. On the positive side, we obtain good exponents for the Erdős-Falconer distance problem for subsets of the unit sphere in F d q and discuss their sharpness. This results in a reasonably complete description of the Erdős-Falconer distance problem in higher-dimensional vector spaces over general finite fields.

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.…”

Quasirandom Group Actions

“…There are a number of comparable sum-product results for large subsets of finite fields; we refer particularly to [11] and to [14,15]. (M. Rudnev has pointed out to me that Proposition 1.7 can be proved in an alternative way, as a consequence of a Szemerédi-Trotter type theorem (for instance [30,Theorem 3]).…”

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.…”