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

TL;DR

This paper demonstrates that in certain thin subgroups of SL(2,Z), almost all admissible primes are represented by integer values in a specific set, with a small exceptional set, using advanced analytic methods.

Contribution

It introduces a novel approach combining the circle method, bilinear forms estimates, and spectral gap results to study integer representations in thin groups.

Findings

Almost all admissible primes are represented in the set S.

The exceptional set of integers not represented has a power savings bound.

The methods extend to thin groups with high Hausdorff dimension.

Abstract

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.