Low-energy decomposition results over finite fields

TL;DR

This paper establishes new low-energy decomposition results for subsets of finite fields, improving bounds on additive and quadratic energy, with applications to expanders, exponential sums, and the Littlewood problem.

Contribution

It introduces novel low-energy decomposition techniques for finite field subsets, extending previous results and providing quantitative bounds with broader applications.

Findings

Decomposition of sets into parts with controlled energy

Improved bounds on additive and quadratic energies

Applications to expanders and exponential sum estimates

Abstract

We prove various low-energy decomposition results, showing that we can decompose a finite set $A\subset \mathbb{F}_p$ satisfying $|A|<p^{5/8}$, into $A = S\sqcup T$ so that, for a non-degenerate quadratic $f\in \mathbb{F}_p[x,y]$, we have \[ |\{(s_1,s_2,s_3,s_4)\in S^4 : s_1 + s_2 = s_3 + s_4\}| \ll |A|^{3 - \frac15 + \varepsilon} \] and \[ |\{(t_1,t_2,t_3,t_4)\in T^4 : f(t_1, t_2) = f(t_3, t_4)\}|\ll |A|^{3 - \frac15 + \varepsilon}\,. \] Variations include extending this result to large $A$ and a low-energy decomposition involving additive energy of images of rational functions. This gives a quantitative improvement to a result of Roche-Newton, Shparlinski and Winterhof as well as a generalisation of a result of Rudnev, Shkredov and Stevens. We consider applications to conditional expanders, exponential sum estimates and the finite field Littlewood problem. In particular, we improve results of Mirzaei, Swaenepoel and Winterhof and Garcia.