A Quantitative Oppenheim Theorem for generic ternary quadratic forms

TL;DR

This paper establishes a quantitative version of Oppenheim's conjecture specifically for generic ternary indefinite quadratic forms, extending recent results from diagonal forms to a broader class.

Contribution

It provides a new quantitative analysis of Oppenheim's conjecture for generic ternary quadratic forms, inspired by Bourgain's work on diagonal forms.

Findings

Proves a quantitative version of Oppenheim's conjecture for generic ternary forms

Extends techniques from diagonal to non-diagonal quadratic forms

Provides bounds and estimates for the distribution of values

Abstract

We prove a quantitative version of Oppenheim's conjecture for generic ternary indefinite quadratic forms. Our results are inspired by and analogous to recent results for diagonal quadratic forms due to Bourgain.