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Uniform congruence counting for Schottky semigroups in SL2(𝐙)

Magee, Michael and Oh, Hee and Winter, Dale (2019) 'Uniform congruence counting for Schottky semigroups in SL2(𝐙).', Journal für die Reine und Angewandte Mathematik. = Crelles journal., 2019 (753). pp. 89-135.


Let Γ be a Schottky semigroup in SL2(Z), and for q∈N, let Γ(q):={γ∈Γ: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.

Item Type:Article
Additional Information:With an appendix by Jean Bourgain, Alex Kontorovich and Michael Magee.
Full text:(AM) Accepted Manuscript
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Publisher statement:The final publication is available at
Date accepted:28 November 2016
Date deposited:07 September 2017
Date of first online publication:12 January 2017
Date first made open access:12 January 2018

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