Some combinatorial number theory problems over finite valuation rings

TL;DR

This paper extends and enhances combinatorial number theory results from finite fields and cyclic rings to the broader context of finite valuation rings, providing new insights and generalizations.

Contribution

It introduces generalizations and improvements of existing combinatorial number theory results over finite valuation rings, broadening their applicability.

Findings

Generalized key results from finite fields to valuation rings

Improved bounds and conditions for combinatorial problems

Unified framework for number theory over valuation rings

Abstract

Let $\mathcal{R}$ be a finite valuation ring of order $q^r$. In this paper we generalize and improve several well-known results, which were studied over finite fields $\mathbb{F}_q$ and finite cyclic rings $\mathbb{Z}/p^r\mathbb{Z}$, in the setting of finite valuation rings.