Rogers' mean value theorem for S-arithmetic Siegel transforms and applications to the geometry of numbers

Abstract

We prove higher moment formulas for Siegel transforms defined over the space of unimodular S-lattices in QSd, d≥3, where in the real case, the formulas are introduced by Rogers [20]. As applications, we obtain the random statements of Gauss circle problem for any star-shaped sets in QSd centered at the origin and of the effective Oppenheim conjecture for S-arithmetic quadratic forms.