Quantitative strong approximation for ternary quadratic forms II

TL;DR

This paper investigates the distribution of rational points on affine quadrics defined by ternary quadratic forms, using advanced circle method techniques to establish equidistribution results related to Linnik's problem.

Contribution

It introduces a $ ext{delta}$-variant of the Hardy--Littlewood circle method to analyze rational points on ternary quadratic forms, advancing understanding of their distribution.

Findings

Rational points are equidistributed in the adelic space off a finite place.

The approach connects to and advances Linnik's problem.

Uses novel $ ext{delta}$-variant of circle method.

Abstract

Let $F$ be a non-degenerate integral ternary quadratic form and let $m_0\in\mathbb{Z}_{\neq 0}$. We study growth of rational points on the affine quadric $(F=m_0)$ and show that they are equidistributed in the adelic space off a finite place. This is closely related to Linnik's problem. Our approach is based on the $\delta$-variant of the Hardy--Littlewood circle method developed by Heath-Brown.