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

4 Oct 2019  ·  Jiyoung Han ·

We prove higher moment formulas for Siegel transforms defined over the space of unimodular $S$-lattices in $\mathbb Q_S^d$, $d\ge 3$, where in the real case, the formulas are introduced by 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...

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Dynamical Systems Number Theory 11H60, 11P21, 37A45