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

2019/10/04 by Jiyoung Han, Han, Jiyoung
Mathematics · #11H60 #11P21 #37A45 #Advanced Algebra and Geometry #Analytic and geometric function theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.DS #math.NT #msc:11H60 #msc:11P21 #msc:37A45

paper · pdf · doi:10.48550/arxiv.1910.01824

27 pages

openalex publication_date 2019/10/04 · openalex created_date 2019/10/10 · arxiv created 2021/05/25 · arxiv updated 2021/05/26 · openalex updated_date 2026/07/28

Abstract

We prove higher moment formulas for Siegel transforms defined over the space of unimodular S-lattices in \mathbb Q_S^d, d≥ 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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