On Representation of Integers from Thin Subgroups of SL(2,Z) with Parabolics

Let $\Lambda<SL(2,\mathbb{Z})$ be a finitely generated, non-elementary Fuchsian group of the second kind, and $v, w$ be two primitive vectors in $\mathbb{Z}^2-(0,0)$. We consider the set $\mathcal{S}=\{\langle {v}\gamma,{w}\rangle_{\mathbb{R}^2}:\gamma\in\Lambda\}$, where $\langle\cdot,\cdot\rangle_{\mathbb{R}^2}$ is the standard inner product in $\mathbb{R}^2$. Using Hardy-Littlewood circle method and some infinite co-volume lattice point counting techniques developed by Bourgain, Kontorovich and Sarnak, together with Gamburd's 5/6 spectral gap, we show that if $\Lambda$ has parabolic elements, and the critical exponent $\delta$ of $\Lambda$ exceeds 0.995371, then a density-one subset of all admissible integers (i.e. integers passing all local obstructions) are actually in $\mathcal{S}$, with a power savings on the size of the exceptional set (i.e. the set of admissible integers failing to appear in $\mathcal{S}$). This supplements a result of Bourgain-Kontorovich, which proves a density-one statement for the case when $\Lambda$ is free, finitely generated, has no parabolics and has critical exponent $\delta>0.999950$.