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 → ∞.