Uniform congruence counting for Schottky semigroups in SL2(𝐙)

Abstract Let Γ be a Schottky semigroup in SL2⁢ (𝐙) SL₂ (Z), and for q∈𝐍 q, let Γ⁢ (q): =γ∈Γ: γ=e⁢ (mod⁢q) (q): =\{: =e~{ (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 BR Bₑ in M2⁢ (𝐑) M₂ (R) of radius R: for all positive integer q with no small prime factors, #⁢ (Γ⁢ (q) ∩BR) =cΓ⁢R2⁢δ#⁢ (SL2⁢ (𝐙/q⁢𝐙) ) +O⁢ (qC⁢R2⁢δ-ϵ) \# ( (q) Bₑ) =c_R^2\# (SL₂ (% Z/qZ) ) +O (q^CR^2-) as R→∞ R for some cΓ>0, C>0, ϵ>0 c_{>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) SL₂ (Z), which arises in the study of Zaremba’s conjecture on continued fractions.