ON REPRESENTATIONS OF INTEGERS IN THIN SUBGROUPS OF SL2(Z)

Let Gamma < SL(2, Z) be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive vectors v(0), w(0) is an element of Z(2) \ {0}. We consider the set S of all integers occurring in < v(0)gamma, w(0)> for gamma is an element of Gamma and the usual inner product on R-2. Assume that the critical exponent delta of Gamma exceeds 0.99995, so that G 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 an element of S (mod q) for all q >= 1) but fail to be globally represented, n is not an element of S, has a power savings, |E(N)| << N1-epsilon 0 for some epsilon(0) > 0, as N -> infinity.