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Balog and Wooley have recently proved that any subset $A$ of either realnumbers 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 somenatural 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 whichleads to the restriction $\# A > q^{1/2}$ where $q$ is the field size. Thebound is optimal, up to logarithmic factors, when $\# A \geq q^{9/13}$. Using$f(X)=X^{-1}$ we apply this result to estimate some triple additive andmultiplicative character sums involving three sets with convolutions $ab+ac+bc$with variables $a,b,c$ running through three arbitrary subsets of a finitefield.

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