Mean Value of $S$-arithmetic Siegel transform: Rogers' mean value theorem for $S$-arithmetic Siegel transform and applications to the geometry of numbers

Jiyoung Han

Published 2019-10-04Version 1

We prove the second moment theorem for Siegel transform defined over the space of unimodular $S$-lattices in $\mathbb Q_S^d$, $d\ge 3$, following the work of Rogers (1955). As applications, we obtain the random statements of Gauss circle problem for any convex sets in $\mathbb Q_S^d$ containing the origin and of the effective Oppenheim conjecture for $S$-arithmetic quadratic forms.