Uniform congruence counting for Schottky semigroups in SL2(𝐙)
AbstractLet Γ be a Schottky semigroup in {\mathrm{SL}_{2}(\mathbf{Z})}, and for {q\in\mathbf{N}}, let{\Gamma(q):=\{\gamma\in\Gamma:\gamma=e~{}(\mathrm{mod}~{}q)\}}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 {B_{R}} in {M_{2}(\mathbf{R})} of radius R: for all positive integer q with no small prime factors,\#(\Gamma(q)\cap B_{R})=c_{\Gamma}\frac{R^{2\delta}}{\#(\mathrm{SL}_{2}(% \mathbf{Z}/q\mathbf{Z}))}+O(q^{C}R^{2\delta-\epsilon})as {R\to\infty} for some {c_{\Gamma}>0,C>0,\epsilon>0} which are independent of q. Our technique also applies to give a similar counting result for the continued fractions semigroup of {\mathrm{SL}_{2}(\mathbf{Z})}, which arises in the study of Zaremba’s conjecture on continued fractions.