Additive energy of polynomial images

Abstract

Given a monic polynomial $f(X)\in {\mathbb{Z}}_m[X]$ over a residue ring ${\mathbb{Z}}_m$ modulo an integer $m\ge 2$ and a discrete interval $\mathcal{I}=\{1,\dots ,H\}$ of $H\le m$ consecutive integers, considered as elements of ${\mathbb{Z}}_m$, we obtain a new upper bound for the additive energy of the set $f(\mathcal{I})$, where $f(\mathcal{I})$ denotes the image set $f(\mathcal{I})=\{f(u):u\in \mathcal{I}\}$. We give an application of our bounds to multiplicative character sums, improving some previous result of Shkredov and Shparlinski~(2018).