Integrability of Siegel transforms and an application

2025/11/19 by Pfitscher, René
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Mathematical Dynamics and Fractals

paper · doi:10.48550/arxiv.2511.15568

Abstract

We establish sharp algebraic criteria for the L^p-integrability, for p = 1, 2, ∞, of a natural generalization of the Siegel transform to the setting of rational representations of semisimple algebraic ℚ-groups, extending Siegel's analytic work in the geometry of numbers. As an application, we derive an effective asymptotic formula for the number of rational approximations of bounded height to almost every real point on a rank-one flag variety at the Diophantine exponent. The argument combines the integrability criterion with effective equidistribution estimates for translated orbits of maximal compact subgroups, a result of independent interest.

Citations

• Counting rational approximations on rank one flag varieties
• Second moment of the light-cone Siegel transform and applications
• Effective rational approximation on spheres
• Groupes arithmétiques et approximation diophantienne
• Quantitative rational approximation on spheres
• Central Limit Theorems for Diophantine approximants
• Expanding cone and applications to homogeneous dynamics
• Intrinsic Diophantine approximation on quadric hypersurfaces
• Rational Approximation on Spheres
• Strong Spectral Gaps for Compact Quotients of Products of \PSL(2,\bbR)

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• Intrinsic Diophantine approximation in Carnot groups and in the Siegel model of the Heisenberg group
• Groupes arithmétiques et approximation diophantienne
• Exceptional Siegel-Weil theorems for compact Spin₈