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

Abstract

Let Λ < SL(2,Z) be a finitely generated, nonelementary Fuchsian group of the 2nd kind, and v,w be two primitive vectors in Z2-0. We consider the set S = {(vγ ,w) R2 : γ ∈Λ}, where ( , ) R2 is the standard inner product in R2. 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 Λ has parabolic elements, and the critical exponent δ of Λ exceeds 0.998317, then a density-one subset of all admissible integers (i.e., integers passing all local obstructions) are actually in S, with a power savings on the size of the exceptional set (i.e., the set of admissible integers failing to appear in S). This supplements a result of Bourgain-Kontorovich, which proves a density-one statement for the case when Λ is free, finitely generated, has no parabolics, and has critical exponent δ > 0.999950. © The Author(s) 2018.