Paper detail

On representations of integers in thin subgroups of SL(2,Z)

Let Gamma < SL(2,Z) be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive non-zero vectors v0, w0 in Z^2. We consider the set S of all integers occurring in <v0 gamma,w0>, for gamma in Gamma. Assume that the limit set of Gamma has Hausdorff dimension delta>0.99995, that is, Gamma 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 |n|<N which are locally admissible (n is in S(mod q) for all q>=1) but fail to be globally represented, n is not in S, has a power savings, $|E(N)| << N^{1-epsilon0}$ for some epsilon>0.