On Representations of Integers in Thin Subgroups of SL₂(ℤ)

Let be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive vectors v₀,w₀ ∈ ℤ². We consider the set S of all integers occurring in 〈 v_{0 γ},w₀ 〉, for γ ∈ and the usual inner product on ℝ. 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 S contains almost all of its admissible primes, that is, those not excluded by local (congruence) obstructions. Moreover, we show that the exceptional set E(N) of integers {pipe}n{pipe} < N which are locally admissible (n ∈ S(mod q) for all q ≥ 1) but fail to be globally represented, n ∉ S, has a power savings, {pipe}E (N){pipe} ≪ N^{1 - 0} for some ₀ > 0, as N → ∞