On values of isotropic quadratic forms

TL;DR

This paper provides asymptotic estimates for the number of solutions to inequalities involving isotropic quadratic forms over certain local fields, using continued fraction expansions of the form's coefficients.

Contribution

It introduces a novel method linking continued fraction expansions to solution counts of quadratic inequalities over local fields.

Findings

Derived asymptotic estimates for solution counts

Connected continued fractions to quadratic form solutions

Applicable to non-discrete, characteristic p>2 fields

Abstract

Let $K$ be a locally compact non-discrete field of characteristic $p>2$ and $Q$ be a non-degenerate isotropic binary quadratic form with coefficients in $K$. We obtain asymptotic estimates for the number of solutions in the two-fold product of a discrete subring inside $K$, of the inequalities of the form $|Q(x,y)|<\delta$ for some $\delta>0$, where $| \cdot |$ is an ultrametric absolute value on $K$. The estimates are obtained in terms of continued fraction expansions of the coefficients of the quadratic form $Q$.