Analogues of the Balog–Wooley Decomposition for Subsets of Finite Fields and Character Sums with Convolutions

Abstract

Balog and Wooley have recently proved that any subset A of either real numbers or of a prime finite field can be decomposed into two parts U and V, one of small additive energy and the other of small multiplicative energy. In the case of arbitrary finite fields, we obtain an analogue that under some natural restrictions for a rational function f both the additive energies of U and f pVq are small. Our method is based on bounds of character sums which leads to the restriction #A ą q 1{2 where q is the field size. The bound is optimal, up to logarithmic factors, when #A ě q 9{13 . Using f pXq " X´1 we apply this result to estimate some triple additive and multiplicative character sums involving three sets with convolutions ab`ac`bc with variables a, b, c running through three arbitrary subsets of a finite field.2010 Mathematics Subject Classification. 11B30, 11T30.

An energy decomposition theorem for matrices and related questions

“…It follows from this theorem that for any set A of matrices in M 2 (F q ), we always can find a subset with either small additive energy or small multiplicative energy. In the setting of finite fields, such a result has many applications in studying exponential sums and other topics, for instance, see [4,7,9,12,13,14,15] and references therein. By the Cauchy-Schwarz inequality, we have the following direct consequence on a sum-product estimate, namely, for A ⊆ GL 2 (F q ), we have…”

“…To prove Theorem 2.1, we will also need several technical results. A proof of the following inequality may be found in [9,Lemma 2.4].…”

“…The authors also proved a finite field variant of this result. Also, see [2,4,9,10] for quantitative improvements, analogues and various applications of these results.…”

Decomposition of subsets of finite fields

“…Our main results are an extension of [4], which themselves are a generalisation of the Balog-Wooley decomposition [1, Theorem 1.3].…”

Additive double character sums over some structured sets and applications

“…implies (5). The condition (6) is in general optimal up to an absolute constant factor (see [7,Section 3.6]).…”