TY - JOUR

T1 - Uniform congruence counting for Schottky semigroups in SL2(Z)

AU - Magee, Michael

AU - Oh, Hee

AU - Winter, Dale

AU - Bourgain, Jean

AU - Kontorovich, Alex

N1 - Funding Information:
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.
Publisher Copyright:
© 2019 De Gruyter.

PY - 2019/8/1

Y1 - 2019/8/1

N2 - Let σ be a Schottky semigroup in SL2.Z/, and for q e N, let be its congruence subsemigroup of level q. Let i denote the Hausdorff dimension of the limit set of σ. 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, as R → ∞ for some cσ > 0; C > 0; > 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.

AB - Let σ be a Schottky semigroup in SL2.Z/, and for q e N, let be its congruence subsemigroup of level q. Let i denote the Hausdorff dimension of the limit set of σ. 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, as R → ∞ for some cσ > 0; C > 0; > 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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U2 - 10.1515/crelle-2016-0072

DO - 10.1515/crelle-2016-0072

M3 - Article

AN - SCOPUS:85048628114

VL - 2019

SP - 89

EP - 135

JO - Journal fur die Reine und Angewandte Mathematik

JF - Journal fur die Reine und Angewandte Mathematik

SN - 0075-4102

IS - 753

ER -