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

Let $\mathrm{\Lambda }<SL(2,\mathbb{Z})$ 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 $$\mathcal{S}$={\langle {v}\gamma,{w}\rangle_{$\mathbb{R}$^2}:\gamma\in\Lambda}$,where$\langle\cdot,\cdot\rangle_{$\mathbb{R}$^2}$isthestandardinnerproductin$$\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 $\mathrm{\Lambda }$ has parabolic elements, and the critical exponent $\delta$ of $\mathrm{\Lambda }$ 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 $\mathrm{\Lambda }$ is free, finitely generated, has no parabolics and has critical exponent $\delta >0.999950$.