Convergence rate for Rényi-type continued fraction expansions (Q2221020)

Let \(f^0_N\in C^1([0,1])\) such that \((f^0_N)'>0\) and let \(\mu\) be a probability measure on \(\mathfrak{B}_{[0,1]}\) such that \(\mu \ll \lambda\). Then the authors prove that for any \(n\in\mathbb{N}_+\) and \(x\in [0,1]\) we have \[\left(\log\left(\frac N{N-1}\right)\right)^2\frac N2 \frac{\alpha_N}{\beta_N} \min_{x\in [0,1]} (f^0_N)'(x) \cdot v_N^n G_N(x)(1-G_N(x)) \leq \] \[\vert \mu (R_N^n<x) -G_N(x)\vert \leq \] \[\left(\log\left(\frac N{N-1}\right)\right)^2\frac N2 \frac{\beta_N}{\alpha_N} \min_{x\in [0,1]} (f^0_N)'(x) \cdot w_N^n G_N(x)(1-G_N(x)) \] where \(\alpha_N\), \(\beta_N\), \(v_N\) and \(w_N\) are suitable constants, \[G_N(x)=\frac 1{\log(\frac N{N-1})}\log\left(\frac {x+N-1}{N-1}\right)\] and \(R_N: [0,1]\to [0,1]\) is the shift transformation defined by \(R_N(1)=0\) and \(R_N([a_1,a_2,a_3,\dots]_R)=[a_2,a_3,\dots]_R\) for \(x\not= 1\). The proof is in the spirit of Wirsing.