On Representations of Integers in Thin Subgroups of SL₂(mathbb Z){{rm SL}_2({mathbb {Z}})}

摘要

Let G < SL(2, mathbb Z){Gamma < {rm SL}(2, {mathbb Z})} be a free, finitely generated Fuchsian group of the second kind with no parabolics, and fix two primitive vectors v₀, w₀ Î mathbb Z2 {0}{v_{0}, w_{0} in mathbb {Z}^{2} , {backslash} , {0}}. We consider the set S{mathcal {S}} of all integers occurring in áv₀g, w₀ñ{langle v_{0}gamma, w_{0}rangle}, for g Î G{gamma in Gamma} and the usual inner product on mathbb R2{mathbb {R}^2}. Assume that the critical exponent δ of Γ exceeds 0.99995, so that Γ is thin but not too thin. Using a variant of the circle method, new bilinear forms estimates and Gamburd’s 5/6-th spectral gap in infinite-volume, we show that S{mathcal {S}} contains almost all of its admissible primes, that is, those not excluded by local (congruence) obstructions. Moreover, we show that the exceptional set mathfrak E(N){mathfrak {E}(N)} of integers |n| < N which are locally admissible (n Î S (mod q) for all q ³ 1){(n in mathcal {S} , , ({rm mod} , q) , , {rm for,all} ,, q geq 1)} but fail to be globally represented, n Ï S{n notin mathcal {S}}, has a power savings, |mathfrak E(N)| 0}, as N → ∞.