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.
All Science Journal Classification (ASJC) codes
- Applied Mathematics