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 $${\mathcal {A}}$$ of either real numbers or of a prime finite field can be decomposed into two parts $${\mathcal {U}}$$ and $${\mathcal {V}}$$ , 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 $${\mathcal {U}}$$ and $$f({\mathcal {V}})$$ are small. Our method is based on bounds of character sums which leads to the restriction $$\# {\mathcal {A}}> q^{1/2}$$ , where q is the field size. The bound is optimal, up to logarithmic factors, when $$\# {\mathcal {A}}\ge q^{9/13}$$ . Using $$f(X)=X^{-1}$$ 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.