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arXiv · 1812.11200

$p$-adic quotient sets II: quadratic forms

Christopher Donnay·Stephan Ramon Garcia·Jeremy Rouse
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Abstract

For $A \subseteq \{1,2,\ldots\}$, we consider $R(A) = \{a/a' : a,a' \in A\}$. If $A$ is the set of nonzero values assumed by a quadratic form, when is $R(A)$ dense in the $p$-adic numbers? We show that for a binary quadratic form $Q$, $R(A)$ is dense in $\mathbb{Q}_{p}$ if and only if the discriminant of $Q$ is a nonzero square in $\mathbb{Q}_{p}$, and for a quadratic form in at least three variables, $R(A)$ is always dense in $\mathbb{Q}_{p}$. This answers a question posed by several authors in 2017.

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Christopher Donnay, Stephan Ramon Garcia, Jeremy Rouse. 2018-12-28. $p$-adic quotient sets II: quadratic forms. https://arxiv.org/abs/1812.11200

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