A quantitative Oppenheim Theorem for generic diagonal quadratic forms

TL;DR

This paper provides a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms, offering power gains and near-optimal results through an analytic number theory approach.

Contribution

It introduces a new quantitative framework for Oppenheim's conjecture applicable to generic forms, with improved bounds and optimality in some cases.

Findings

Established power gains in the quantitative bounds

Achieved near-optimal results for certain forms

Applied analytic number theory techniques effectively

Abstract

We establish a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms using an analytic number theory approach. The statements come with power gains and in some cases are essentially optimal