On the Gauss-Kuzmin-Lévy problem for nearest integer continued fractions

Abstract

This note provides an effective bound in the Gauss-Kuzmin-Lévy problem for some Gauss type shifts associated with nearest integer continued fractions, acting on the interval \(I_0=\left[ 0,\frac{1}{2}\right] \) or \(I_0=\left[ -\frac{1}{2},\frac{1}{2}\right] \). We prove asymptotic formulas \(\lambda (T^{-n}I) =\mu (I)(\lambda ( I_0) +O(q^n))\) for such transformations T, where \(\lambda \) is the Lebesgue measure on \({\mathbb {R}}\), \(\mu \) the normalized T-invariant Lebesgue absolutely continuous measure, I subinterval in \(I_0\), and \(q=0.288\) is smaller than the Wirsing constant \(q_W\approx 0.3036\).