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Discrepancy of rational points in simple algebraic groups
Part of:
Discontinuous groups and automorphic forms
Lie groups
Diophantine approximation, transcendental number theory
Ergodic theory
Published online by Cambridge University Press: 13 March 2024
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.
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- © 2024 The Author(s). The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence
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