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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)



Quantitative ergodic theorems and their number-theoretic applications

Authors: Alexander Gorodnik and Amos Nevo
Journal: Bull. Amer. Math. Soc. 52 (2015), 65-113
MSC (2010): Primary 37A15, 37P55, 22E46, 11J83, 11F70
Published electronically: June 11, 2014
MathSciNet review: 3286482
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Abstract: We present an account of some recent applications of ergodic theorems for actions of algebraic and arithmetic groups to the solution of natural problems in Diophantine approximation and number theory. Our approach is based on spectral methods utilizing the unitary representation theory of the groups involved. This allows the derivation of ergodic theorems with a rate of convergence, an important phenomenon which does not arise in classical ergodic theory. Combining spectral and dynamical methods, quantitative ergodic theorems give rise to new and previously inaccessible applications. We demonstrate the remarkable diversity of such applications by deriving general uniform error estimates in non-Euclidean lattice points counting problems, explicit estimates in the sifting problem for almost-prime points on symmetric varieties, best-possible bounds for exponents of intrinsic Diophantine approximation on homogeneous algebraic varieties, and quantitative results on fast distribution of dense orbits on compact and non-compact homogeneous spaces.

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Additional Information

Alexander Gorodnik
Affiliation: School of Mathematics, University of Bristol, Bristol, United Kingdom

Amos Nevo
Affiliation: Department of Mathematics, Technion, Israel

Received by editor(s): April 25, 2013
Published electronically: June 11, 2014
Additional Notes: The first author was supported in part by EPSRC, ERC, and RCUK
The second author was supported by an ISF grant
Article copyright: © Copyright 2014 American Mathematical Society