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

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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 \(I_0=\left[ 0,\frac{1}{2}\right] \) 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 \(q_W\approx 0.3036\).

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Acknowledgements

This research was supported by NSF award DMS-1449269 and University of Illinois Research Board Award RB-22069. We are grateful to Joseph Vandehey for constructive comments and to the referees for careful reading and valuable suggestions.

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Author notes

  1. Maria Siskaki

    Present address: Department of Mathematics, Yale University, New Haven, CT, 06511, USA

Authors and Affiliations

  1. Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL, 61801, USA

    Florin P. Boca & Maria Siskaki

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  1. Florin P. Boca
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Correspondence to Florin P. Boca.

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Communicated by H. Bruin.

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Boca, F.P., Siskaki, M. On the Gauss-Kuzmin-Lévy problem for nearest integer continued fractions. Monatsh Math (2024). https://doi.org/10.1007/s00605-024-01968-w

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