On $p$-adic denseness of quotients of values of integral forms

Abstract: Given $A\subseteq \mathbb{Z}$, the ratio set or the quotient set of $A$ is defined by $R(A):={a/b: a, b\in A, b\neq 0}$. It is an open problem to study the denseness of $R(A)$ in the $p$-adic numbers when $A$ is the set of values attained by an integral form. In this paper we give a sufficient condition for the existence of infinitely many primes $p$ such that the ratio set of the values of an integral form is dense in $\mathbb{Q}_p$. We prove that if a form is non-singular and has at least three variables, then the ratio set of its values is dense in $\mathbb{Q}_p$ for all sufficiently large $p$. We give two counterexamples to this statement if we do not assume non-singularity. The first one happens when the number of variables is equal to the degree. The other works for any number of variables but only for composite degrees, which motivates our conjecture that there are no such counterexamples of prime degree. Finally, we construct integral forms such that the ratio set of its values is dense in at least one $\mathbb{Q}_p$ but only in finitely many of them.