P-adic quotient sets: cubic forms

Abstract

For \(A\subseteq \{1, 2, \ldots \}\), we consider \(R(A)=\{a/b: a, b\in A\}\). It is an open problem to study the denseness of R(A) in the p-adic numbers when A is the set of nonzero values assumed by a cubic form. We study this problem for the cubic forms \(ax^3+by^3\), where a and b are integers. We also prove that if A is the set of nonzero values assumed by a non-degenerate, integral, and primitive cubic form with more than 9 variables, then R(A) is dense in \({\mathbb {Q}}_p\).