$p$-adic quotient sets

TL;DR

This paper explores the density of quotient sets in the $p$-adic numbers, extending classical real number results and employing various number theory techniques to identify conditions for density.

Contribution

It introduces the study of quotient set density in the $p$-adic context, an area largely unexplored compared to the real numbers, and develops new conditions using diverse number theory methods.

Findings

Identifies conditions for $R(A)$ to be dense in $Q_p$

Employs algebraic and analytic number theory techniques

Poses open questions for future research

Abstract

For $A \subseteq \mathbb{N}$, the question of when $R(A) = \{a/a' : a, a' \in A\}$ is dense in the positive real numbers $\mathbb{R}_+$ has been examined by many authors over the years. In contrast, the $p$-adic setting is largely unexplored. We investigate conditions under which $R(A)$ is dense in the $p$-adic numbers. Techniques from elementary, algebraic, and analytic number theory are employed in this endeavor. We also pose many open questions that should be of general interest.