Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Uniform congruence counting for Schottky semigroups in
  $\mathrm{SL}_2(\mathbf{Z})$   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.

24
The categories of this paper are: dg-ga
12
The categories of this paper are: nucl-th
33
The categories of this paper are: plasm-ph
26
The categories of this paper are: bayes-an
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
4
The categories of this paper are: cond-mat
35
The categories of this paper are: mtrl-th
16
The categories of this paper are: stat
20
The categories of this paper are: acc-phys
23
The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
8
The categories of this paper are: hep-ex
25
The categories of this paper are: patt-sol
5
The categories of this paper are: gr-qc
31
The categories of this paper are: ao-sci