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arXiv · 2209.07452

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

Florin P. Boca·Maria Siskaki
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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 $I_0=[0,\frac{1}{2}]$ or $I_0=[-\frac{1}{2},\frac{1}{2}]$. We prove asymptotic formulas $λ(T^{-n}I) =μ(I)(\vert I_0 \vert +O(q^n))$ for such transformations $T$, where $λ$ is the Lebesgue measure on $\mathbb R$, $μ$ the normalized $T$-invariant Lebesgue absolutely continuous measure, $I$ subinterval in $I_0$, and $q=0.288$ is smaller than the Wirsing constant $q_W=0.3036\ldots$

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Florin P. Boca, Maria Siskaki. 2024-04-01. On the Gauss-Kuzmin-Lévy problem for nearest integer continued fractions. https://arxiv.org/abs/2209.07452

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