Quantitative strong approximation for ternary quadratic forms I

Zhizhong Huang

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arXiv · 2412.03350

Quantitative strong approximation for ternary quadratic forms I

Zhizhong Huang

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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.

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Zhizhong Huang. 2024-12-04. Quantitative strong approximation for ternary quadratic forms I. https://arxiv.org/abs/2412.03350

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