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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

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Mathematical Citation Quotient (MCQ) 2017: 1.49

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1435-5345
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Volume 2019, Issue 753

Issues

Uniform congruence counting for Schottky semigroups in SL2(𝐙)

Michael Magee / Hee Oh
  • Mathematics Department, Yale University, New Haven, CT 06511, USA; and Korea Institute for Advanced Study, Seoul, South Korea
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/ Dale Winter
Published Online: 2017-01-12 | DOI: https://doi.org/10.1515/crelle-2016-0072

Abstract

Let Γ be a Schottky semigroup in SL2(𝐙), and for q𝐍, let

Γ(q):={γΓ:γ=e(modq)}

be its congruence subsemigroup of level q. Let δ denote the Hausdorff dimension of the limit set of Γ. We prove the following uniform congruence counting theorem with respect to the family of Euclidean norm balls BR in M2(𝐑) of radius R: for all positive integer q with no small prime factors,

#(Γ(q)BR)=cΓR2δ#(SL2(𝐙/q𝐙))+O(qCR2δ-ϵ)

as R for some cΓ>0,C>0,ϵ>0 which are independent of q. Our technique also applies to give a similar counting result for the continued fractions semigroup of SL2(𝐙), which arises in the study of Zaremba’s conjecture on continued fractions.

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About the article

With an appendix by Jean Bourgain at Princeton, Alex Kontorovich at New Brunswick and Michael Magee at New Haven.


Received: 2016-02-01

Revised: 2016-11-28

Published Online: 2017-01-12

Published in Print: 2019-08-01


Bourgain is supported in part by NSF grant DMS-1301619. Kontorovich is supported in part by an NSF CAREER grant DMS-1254788 and DMS-1455705, an NSF FRG grant DMS-1463940, and Alfred P. Sloan Research Fellowship, and a BSF grant. Magee was supported in part by NSF Grant DMS-1128155. Oh was supported in part by NSF Grant DMS-1361673.


Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2019, Issue 753, Pages 89–135, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: https://doi.org/10.1515/crelle-2016-0072.

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