Erratum to: On Representations of Integers in Thin Subgroups of $${{\rm {SL}}_2(\mathbb {Z})}$$

Abstract

Let Γ < SL(2,Z) be finitely generated, free, with no parabolic elements, and having critical exponent 1/2 < δ < 1. As such, there exists a finite symmetric set A := {A1, A−1 1 , A2, A−1 2 , . . . , Ak, A−1 k } of generators with no relations, so that every γ ∈ Γ is expressed uniquely as some reduced word γ = B1B2 · · ·Bm with Bj ∈ A. (Reduced means no annihilations, BjBj+1 = I). Mimicing [BK, §3, (3.2)], we construct the following exponential sum. Fix N 1 and 0 < σ < 1/4. Let Ξ be a subset of Γ containing elements of Frobenius norm at most N1/2, such that all elements ξ ∈ Ξ, when written as a reduced word in the generators of Γ, start with the same letter. Similarly, let Π be a subset of Γ containing elements of Frobenius norm at most N1/2−σ, and all elements ∈ Π, when written as a reduced word in the generators of Γ, ending in the same letter (ensuring that it is not the inverse of the letter which starts all elements of Ξ). Then for θ ∈ [0, 1], and primitive v0, w0 ∈ Z2, let SN (θ) := ∑