Generalised Fermat equations in dense variables over finite fields and rings

TL;DR

This paper demonstrates that generalized Fermat equations have the expected number of solutions over dense subsets of finite fields and rings, establishing optimal density thresholds using advanced harmonic analysis and combinatorial techniques.

Contribution

It introduces new bounds and methods for analyzing solutions to Fermat equations over dense subsets of finite structures, extending previous results with optimal density conditions.

Findings

Solutions exist in expected numbers over dense subsets

Density thresholds are proven to be optimal

Advanced harmonic and combinatorial techniques are employed

Abstract

Let $A$ be a sufficiently dense subset of a finite field $\mathbb F_q$ or a finite, cyclic ring $\mathbb Z/ N\mathbb Z$. Assuming that $q$ and $N$ have no small prime divisors, we show that generalised Fermat equations have the expected number of solutions over $A$. We further show that our density threshold is optimal. Our proofs involve average Fourier decay for Bohr sets, mixed character sum bounds, equidistribution of polynomial sequences, popular Cauchy--Davenport lemmas, and a regularity-type lemma due to Semchankau.