$p$-adic quotient sets II: quadratic forms (1812.11200v2)

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.