A note on $p$-adic denseness of quotients of values of quadratic forms

TL;DR

This paper provides an alternative proof for the classification of quadratic forms with integral coefficients whose quotients of values are dense in the $p$-adic numbers, building on prior work by Donnay, Garcia, and Rouse.

Contribution

It offers a new proof of the existing classification of quadratic forms with dense quotients in $p$-adic fields, enhancing understanding of their properties.

Findings

Classification of quadratic forms with dense $p$-adic quotients confirmed

New proof technique for the classification presented

Supports previous results with alternative approach

Abstract

Donnay, Garcia and Rouse classified nonsingular quadratic forms $Q$ with integral coefficients and prime numbers $p$ such that the set of quotients of values of $Q$ attained for integer arguments is dense in the field of $p$-adic numbers. The aim of this note is to give another proof of this classification.