Arithmetic purity of the Hardy-Littlewood property and geometric sieve for affine quadrics

We establish the Hardy-Littlewood property (`a la Borovoi-Rudnick) for Zariski open subsets in affine quadrics of the form $q(x_1,\cdots ,x_n)=m$, where $q$ is a non-degenerate integral quadratic form in $n⩾3$ variables and $m$ is a non-zero integer. This gives asymptotic formulas for the density of integral points taking coprime polynomial values, which is a quantitative version of the arithmetic purity of strong approximation property off infinity for affine quadrics.