$p$-adic quotient sets: Cubic forms

Abstract

For $A\subseteq \{1,2,\dots \}$, 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 $a{x}^3+b{y}^3$, 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 ${\mathbb{Q}}_p$.

6 pages