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

Abstract

Abstract Let $\Lambda <SL(2,\mathbb{Z})$ be a finitely generated, nonelementary Fuchsian group of the 2nd kind, and $\mathbf{v},\mathbf{w}$ be two primitive vectors in $\mathbb{Z}^2\!-\!\mathbf{0}$. We consider the set $\mathcal{S}\!=\!\{\left \langle \mathbf{v}\gamma ,\mathbf{w}\right \rangle _{\mathbb{R}^2}\!:\!\gamma\! \in\! \Lambda \}$, where $\left \langle \cdot ,\cdot \right \rangle _{\mathbb{R}^2}$ 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 0.998317, 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 $\delta>0.999950$.