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

TL;DR

This paper proves that for certain thin subgroups of SL(2,Z) with parabolic elements and high enough critical exponent, almost all admissible integers can be represented by vectors associated with the group, using advanced number theory techniques.

Contribution

It extends previous results by including groups with parabolic elements and slightly lower critical exponent thresholds, demonstrating density-one representation of integers.

Findings

Density-one of admissible integers are represented in the set for groups with parabolics.

Power savings on the exceptional set size.

Applicable to groups with critical exponent exceeding 0.995371.

Abstract

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 $\delta>0.999950$.