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

Abstract Let 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\!=\!\ v, w ₑℂ\!: \!\! \! \, where, ₑℂ 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 has parabolic elements, and the critical exponent of 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 is free, finitely generated, has no parabolics, and has critical exponent 0. 999950.