Effective equidistribution, arithmetic purity of strong approximation, and geometric sieve for affine quadrics

Abstract

Let $k$ be a number field. Let $q(x_1,\cdots ,x_n)$ be a non-degenerate integral quadratic form in $n\ge 3$ variables with coefficients in $k$ and $m\in k^{\times }$. Let $X$ be the affine quadric defined by $q=m$ in ${\mathbb{A}}_k^n$. Based on results on effective equidistribution of $S$-integral points in symmetric spaces, we establish the following: (i) The arithmetic purity of strong approximation off any single place of $k$ for $X$; (ii) The geometric sieve for $p_0$-integral points on $X$ when $k=\mathbb{Q}$ and $p_0$ is a prime number.