On Gauss problem for the Lüroth expansion

Consider the transformation τ(x)=[1x](([1x]+1)x−1),x≠0,τ(0)=0, of I=[0,1] which underlines the Lüroth expansion. Let μ≪λ (Lebesgue measure on [0,1]). We show that μτ−n(A) approaches λ(A) uniformly in A∈BI with reminder O(qn),0