Decomposition of subsets of finite fields

Abstract

We extend a bound of Roche-Newton, Shparlinski and Winterhof which says any subset of a finite field can be decomposed into two disjoint subset $\mathcal{U}$ and $\mathcal{V}$ of which the additive energy of $\mathcal{U}$ and $f(\mathcal{V})$ are small, for suitably chosen rational functions $f$. We extend the result by proving equivalent results over multiplicative energy and the additive and multiplicative energy hybrids.