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

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 \(I_0=\left[ 0,\frac{1}{2}\right] \) $I_0=\left[0,,,\frac{1}{2}\right]$ or \(I_0=\left[ -\frac{1}{2},\frac{1}{2}\right] \) $I_0=\left[-,\frac{1}{2},,,\frac{1}{2}\right]$ . We prove asymptotic formulas \(\lambda (T^{-n}I) =\mu (I)(\lambda ( I_0) +O(q^n))\) $\lambda (T^{-n}I)=\mu (I)(\lambda (I_0)+O(q^n))$ for such transformations T, where \(\lambda \) $\lambda$ is the Lebesgue measure on \({\mathbb {R}}\) $\mathbb{R}$ , \(\mu \) $\mu$ the normalized T-invariant Lebesgue absolutely continuous measure, I subinterval in \(I_0\) $I_0$ , and \(q=0.288\) $q=0.288$ is smaller than the Wirsing constant \(q_W\approx 0.3036\) $q_W\approx 0.3036$ .