Additive Combinatorics over Finite Fields: New Results and Applications

Abstract

We give a survey of recently emerged directions in additive combinatorics over finite fields. We describe a variety of concrete results and outline their applications to a broad spectrum of other problems.

Elements of Large Order in Prime Finite Fields

“…This question is also notoriously difficult and there is an extensive literature with various contributions. This note is mainly motivated by a paper of Voloch [V1] and earlier work of von zur Gathen and Shparlinski [GS,S1]. The main result in [V1] states roughly that if F(x, y) ∈ F q [x, y] is absolutely irreducible and F(x, 0) is not a monomial, given a solution (a, b) ∈ F * q × F * q of F(x, y) = 0 such that d = [F q (a) : F q ] is sufficiently large, then either a is of multiplicative order at least d 2− or b is of order at least exp(δ(log d) 2 ).…”

Spherical Configurations over Finite Fields

“…The setting of vector spaces over finite fields provides a useful model to study many problems in arithmetic combinatorics; see especially the surveys [14,25,28]. In the context of geometric Ramsey theory over finite fields, notable results have been obtained by a number of authors [3,15,16,24].…”

Additive Combinatorics: With a View Towards Computer Science and Cryptography—An Exposition

“…See [321] for a book on additive combinatorics, [237,238] for two books on additive number theory, and [330,339] for two surveys on additive combinatorics. About additive combinatorics over finite fields and its applications, the reader is referred to the very recent and excellent survey by Shparlinski [291].…”