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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie


IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

SCImago Journal Rank (SJR) 2018: 2.554
Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

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1435-5345
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Volume 2012, Issue 663

Issues

Counting lattice points

Alexander Gorodnik / Amos Nevo
Published Online: 2011-06-17 | DOI: https://doi.org/10.1515/CRELLE.2011.096

Abstract

For a locally compact second countable group G and a lattice subgroup Γ, we give an explicit quantitative solution of the lattice point counting problem in general domains in G, provided that

  1. G has finite upper local dimension, and the domains satisfy a basic regularity condition,

  2. the mean ergodic theorem for the action of G on G/Γ holds, with a rate of convergence.

The error term we establish matches the best current result for balls in symmetric spaces of simple higher-rank Lie groups, but holds in much greater generality.

A significant advantage of the ergodic theoretic approach we use is that the solution to the lattice point counting problem is uniform over families of lattice subgroups provided they admit a uniform spectral gap. In particular, the uniformity property holds for families of finite index subgroups satisfying a quantitative variant of property τ.

We discuss a number of applications, including: counting lattice points in general domains in semisimple S-algebraic groups, counting rational points on group varieties with respect to a height function, and quantitative angular (or conical) equidistribution of lattice points in symmetric spaces and in affine symmetric varieties.

We note that the mean ergodic theorems which we establish are based on spectral methods, including the spectral transfer principle and the Kunze–Stein phenomenon. We formulate and prove appropriate analogues of both of these results in the set-up of adele groups, and they constitute a necessary ingredient in our proof of quantitative results for counting rational points.

About the article

Received: 2009-02-19

Revised: 2010-06-22

Published Online: 2011-06-17

Published in Print: 2012-02-01


Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2012, Issue 663, Pages 127–176, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/CRELLE.2011.096.

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[1]
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[2]
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[3]
R. Vehkalahti, Hsiao-Feng Lu, and L. Luzzi
IEEE Transactions on Information Theory, 2013, Volume 59, Number 9, Page 6060
[4]
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Algebra & Number Theory, 2014, Volume 8, Number 4, Page 999

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