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 $I_0=[0,\frac{1}{2}]$ or $I_0=[-\frac{1}{2},\frac{1}{2}]$. We prove asymptotic formulas $\lambda (T^{-n}I)=\mu (I)(\mathrm{\mid }{I}_0\mathrm{\mid }+O(q^n))$ 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 $q_W=0.3036\dots$