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Continued fraction expansions with variable numerators

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Abstract

A new continued fraction expansion algorithm, the so-called \(a/b\)-expansion, is introduced and some of its basic properties, such as convergence of the algorithm and ergodicity of the underlying dynamical system, have been obtained. Although seemingly a minor variation of the regular continued fraction (RCF) expansion and its many variants (such as Nakada’s \(\alpha \)-expansions, Schweiger’s odd- and even-continued fraction expansions, and the Rosen fractions), these \(a/b\)-expansions behave very differently from the RCF and many important question remains open, such as the exact form of the invariant measure, and the “shape” of the natural extension.

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Notes

  1. All almost sure statements in this paper are wrt. Lebesgue measure \(\lambda \).

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Authors and Affiliations

  1. Department of Mathematics, Utrecht University, Postbus 80.000, 3508 TA , Utrecht, The Netherlands

    Karma Dajani

  2. Delft University of Technology, EWI (DIAM), Mekelweg 4, 2628 CD , Delft, The Netherlands

    Cor Kraaikamp & Niels Daniël Simon Langeveld

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  1. Karma Dajani
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  2. Cor Kraaikamp
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  3. Niels Daniël Simon Langeveld
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Correspondence to Cor Kraaikamp.

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Dajani, K., Kraaikamp, C. & Langeveld, N.D.S. Continued fraction expansions with variable numerators. Ramanujan J 37, 617–639 (2015). https://doi.org/10.1007/s11139-014-9569-4

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