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.
Categories: math.NT