Generalised Fermat equations in dense variables over finite fields and rings

Let $A$ be a sufficiently dense subset of a finite field ${\mathbb{F}}_q$ or a finite, cyclic ring $\mathbb{Z}\mathrm{/}N\mathbb{Z}$. 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.