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arXiv · 1607.07951

$p$-adic quotient sets

Stephan Ramon Garcia·Yu Xuan Hong·Florian Luca·Elena Pinsker·Carlo Sanna·Evan Schechter·Adam Starr
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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.

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Stephan Ramon Garcia, Yu Xuan Hong, Florian Luca, Elena Pinsker, Carlo Sanna, Evan Schechter, Adam Starr. 2016-07-27. $p$-adic quotient sets. https://arxiv.org/abs/1607.07951

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