Decomposition of subsets in finite fields

TL;DR

This paper generalizes a decomposition method for subsets of finite fields, providing bounds on additive and multiplicative energies for various functions and their hybrids, enhancing understanding of subset structures.

Contribution

It extends previous bounds by including multiplicative and hybrid energy decompositions in finite fields, broadening the scope of subset analysis.

Findings

Decomposition bounds for additive energy

Extension to multiplicative energy

Hybrid energy decomposition results

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 $\cU$ and $\cV$ of which the additive energy of $\cU$ and $f(\cV)$ 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.