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Generalised Fermat equations in dense variables over finite fields and rings

Sam Chow, Zi Li Lim, Akshat Mudgal

TL;DR

This work develops a density-regularization framework for generalised Fermat equations in dense settings over both finite fields and finite cyclic rings. It introduces wrappers based on unions of inhomogeneous Bohr sets and deploys Semchankau's wrapping technique, Green--Ruzsa-style popular bounds, and Deligne-type exponential-sum estimates to obtain optimal density thresholds and supersaturation results. The method combines equidistribution of polynomial sequences with robust Fourier-analytic decay in wrappers, extending from finite fields to rings via a product-structure analysis of $\mathbb Z/N\mathbb Z$. The results yield sharp density thresholds and quantitative supersaturation, with potential implications for dense Waring-type problems in these algebraic settings.

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.

Generalised Fermat equations in dense variables over finite fields and rings

TL;DR

This work develops a density-regularization framework for generalised Fermat equations in dense settings over both finite fields and finite cyclic rings. It introduces wrappers based on unions of inhomogeneous Bohr sets and deploys Semchankau's wrapping technique, Green--Ruzsa-style popular bounds, and Deligne-type exponential-sum estimates to obtain optimal density thresholds and supersaturation results. The method combines equidistribution of polynomial sequences with robust Fourier-analytic decay in wrappers, extending from finite fields to rings via a product-structure analysis of . The results yield sharp density thresholds and quantitative supersaturation, with potential implications for dense Waring-type problems in these algebraic settings.

Abstract

Let be a sufficiently dense subset of a finite field or a finite, cyclic ring . Assuming that and have no small prime divisors, we show that generalised Fermat equations have the expected number of solutions over . 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.
Paper Structure (13 sections, 16 theorems, 103 equations)

This paper contains 13 sections, 16 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wrappers
  4. A popular Cauchy--Davenport lemma
  5. Deligne's bound
  6. Semchankau's wrapping lemma
  7. Finite fields
  8. Equidistribution of powers
  9. Average Fourier decay
  10. Dense Fermat over finite fields
  11. Finite, cyclic rings
  12. Equidistribution of powers in wrappers
  13. Dense Fermat over finite, cyclic rings

Key Result

Corollary 1.1

For every $\kappa >0$, every prime $p$ sufficiently large in terms of $\kappa$, and every set $A \subseteq \mathbb{F}_p$ satisfying $|A| \geqslant p(1/8 + \kappa)$, one has

Theorems & Definitions (18)

  • Corollary 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Example 1.6
  • Example 1.7
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 8 more