Uniform congruence counting for Schottky semigroups in $\mathrm{SL}_2(\mathbf{Z})$

TL;DR

This paper establishes a uniform congruence counting theorem for Schottky semigroups in SL(2,Z), providing asymptotic counts with error bounds that are independent of the level q, and applies to continued fractions semigroups related to Zaremba's conjecture.

Contribution

It introduces a uniform counting method for Schottky semigroups in SL(2,Z) that works uniformly across levels q, including applications to continued fractions.

Findings

Proved a uniform asymptotic counting formula for Schottky semigroups.

Established error bounds independent of the level q.

Extended results to continued fractions semigroups related to Zaremba's conjecture.

Abstract

Let $\Gamma$ be a Schottky semigroup in $\mathrm{SL}_2(\mathbf{Z})$, and for $q\in \mathbf N$, let $\Gamma(q):=\{\gamma\in \Gamma: \gamma= e \text{ (mod $q$)}\}$ be its congruence subsemigroup of level $q$. 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 $q$ 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.