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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

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

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Quantitative ergodic theorems and their number-theoretic applications
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by Alexander Gorodnik and Amos Nevo PDF
Bull. Amer. Math. Soc. 52 (2015), 65-113 Request permission


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
  • Email:
  • Amos Nevo
  • Affiliation: Department of Mathematics, Technion, Israel
  • Email:
  • 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
  • © Copyright 2014 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 52 (2015), 65-113
  • MSC (2010): Primary 37A15, 37P55, 22E46, 11J83, 11F70
  • DOI:
  • MathSciNet review: 3286482