Quantitative strong approximation for ternary quadratic forms I

TL;DR

This paper develops an asymptotic counting formula with a secondary term for solutions to ternary quadratic equations, extending understanding of their distribution under local conditions using advanced analytic methods.

Contribution

It introduces a new asymptotic formula with a secondary term for counting solutions to indefinite ternary quadratic forms, employing the δ-variant of the Hardy-Littlewood circle method.

Findings

Derived asymptotic formulas with secondary terms for solution counts

Extended results to forms with growing parameter m

Applied δ-variant of Hardy-Littlewood circle method

Abstract

We derive asymptotic formulas with a secondary term for the (smoothly weighted) count of number of integer solutions of height $\leqslant B$ with local conditions to the equation $F(x_1,x_2,x_3)=m$, where $F$ is a non-degenerate indefinite ternary integral quadratic form, and $m$ is a non-zero integer satisfying $-m\Delta_F=\square$ which can grow like $O(B^{2-\theta})$ for some fixed $\theta>0$. Our approach is based on the $\delta$-variant of the Hardy--Littlewood circle method developed by Heath-Brown.