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

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(V) are small. Our method is based on bounds of character sums which leads to the restriction #A>q1/2, where q is the field size. The bound is optimal, up to logarithmic factors, when #Aq9/13. 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.