$p$-adic quotient sets II: quadratic forms

Abstract

For $A\subseteq \{1,2,\dots \}$, we consider $R(A)=\{a/a^{\prime }:a,a^{\prime }\in A\}$. 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.

14 pages