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arXiv · 2601.00135

Generalised Fermat equations in dense variables over finite fields and rings

Sam Chow·Zi Li Lim·Akshat Mudgal
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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.

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Sam Chow, Zi Li Lim, Akshat Mudgal. 2025-12-31. Generalised Fermat equations in dense variables over finite fields and rings. https://arxiv.org/abs/2601.00135

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