On Representations of Integers in Thin Subgroups of ${{\rm SL}_2({\mathbb {Z}})}$

Abstract

Let ${\Gamma < {\rm SL}(2, {\mathbb Z})}$ be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive vectors ${v_{0}, w_{0} \in \mathbb {Z}^{2} \, {\backslash} \, \{0\}}$. We consider the set ${\mathcal {S}}$ of all integers occurring in ${\langle v_{0}\gamma, w_{0}\rangle}$, for ${\gamma \in \Gamma}$ and the usual inner product on ${\mathbb {R}^2}$. Assume that the critical exponent δ of Γ exceeds 0.99995, so that Γ is thin but not too thin. Using a variant of the circle method, new bilinear forms estimates and Gamburd’s 5/6-th spectral gap in infinite-volume, we show that ${\mathcal {S}}$ contains almost all of its admissible primes, that is, those not excluded by local (congruence) obstructions. Moreover, we show that the exceptional set ${\mathfrak {E}(N)}$ of integers |n| < N which are locally admissible ${(n \in \mathcal {S} \, \, ({\rm mod} \, q) \, \, {\rm for\,all} \,\, q \geq 1)}$ but fail to be globally represented, ${n \notin \mathcal {S}}$, has a power savings, ${|\mathfrak {E}(N)| \ll N^{1-\varepsilon_{0}}}$ for some ${\varepsilon_{0} > 0}$, as N → ∞.