Uniform congruence counting for Schottky semigroups in SL2(Z)

Michael Magee, Hee Oh, Dale Winter, Jean Bourgain, Alex Kontorovich

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


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.

Original languageEnglish (US)
Pages (from-to)89-135
Number of pages47
JournalJournal fur die Reine und Angewandte Mathematik
Issue number753
StatePublished - Aug 1 2019

All Science Journal Classification (ASJC) codes

  • Mathematics(all)
  • Applied Mathematics


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