On S-Hardy-Littlewood homogeneous spaces

We study the asymptotic distribution of S-integral points on affine homogeneous spaces in the light of the Hardy-Littlewood property introduced by Borovoi and Rudnick. We introduce the S-Hardy-Littlewood property for affine homogeneous spaces defined over an algebraic number field and a finite set S of places of the base field. We work with the adelic harmonic analysis on affine algebraic groups over a number field to determine the asymptotic density of S-integral points under congruence conditions. We give some new examples of strongly or relatively S-Hardy-Littlewood homogeneous spaces over number fields. As an application, we prove certain asymptotically uniform distribution property of integral points on an ellipsoid defined by a totally positive definite tenary quadratic form over a totally real number field.