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Discrepancy of rational points in simple algebraic groups

Part of: Discontinuous groups and automorphic forms Ergodic theory Lie groups Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  13 March 2024

Alexander Gorodnik  and
Amos Nevo
Alexander Gorodnik
Affiliation:
Institute für Mathematik, Universität Zürich, CH-8057 Zürich, Switzerland alexander.gorodnik@math.uzh.ch
Amos Nevo
Affiliation:
Faculty of Mathematics Technion, Israel Institute of Technology, 32000 Haifa, Israel amosnevo6@gmail.com and Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
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Abstract

The aim of the present paper is to derive effective discrepancy estimates for the distribution of rational points on general semisimple algebraic group varieties, in general families of subsets and at arbitrarily small scales. We establish mean-square, almost sure and uniform estimates for the discrepancy with explicit error bounds. We also prove an analogue of W. Schmidt's theorem, which establishes effective almost sure asymptotic counting of rational solutions to Diophantine inequalities in the Euclidean space. We formulate and prove a version of it for rational points on the group variety, with an effective bound which in some instances can be expected to be the best possible.

Keywords

semisimple groupsarithmetic lattice subgroupsDiophantine inequalitiesautomorphic representations

MSC classification

Primary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Secondary: 37A17: Homogeneous flows 22E46: Semisimple Lie groups and their representations 11J83: Metric theory
Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 4 , April 2024 , pp. 836 - 877
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Copyright
© 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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