$p$-adic quotient sets

Abstract

For $A\subseteq \mathbb{N}$, the question of when $R(A)=\{a/a^{\prime }:a,a^{\prime }\in A\}$ 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.

24 pages