$p$-adic quotient sets: Cubic forms

TL;DR

This paper investigates the density of quotient sets generated by nonzero values of cubic forms in the p-adic numbers, proving density results for forms with many variables and specific forms in two variables.

Contribution

It extends the understanding of p-adic density of quotient sets to cubic forms, especially for forms with more than 9 variables and specific binary forms.

Findings

Density of R(A) for forms with >9 variables

Density results for binary forms ax^3+by^3

Open problem addressed for specific 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 $\mathbb{Q}_p$.