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

Abstract

We establish the Hardy-Littlewood property (à la Borovoi-Rudnick) for Zariski open subsets in affine quadrics of the form q(x1,⋯,xn)=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.