## $p$-adic quotient sets

### Volume 179 / 2017

Acta Arithmetica 179 (2017), 163-184
MSC: Primary 11B05; Secondary 11A07, 11B39.
DOI: 10.4064/aa8579-4-2017
Published online: 16 June 2017

#### 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. We also pose many open questions that should be of general interest.