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A natural probability measure derived from Stern’s diatomic sequence

Volume 183 / 2018

Michael Baake, Michael Coons Acta Arithmetica 183 (2018), 87-99 MSC: Primary 11B85; Secondary 42A38, 28A80. DOI: 10.4064/aa170709-22-1 Published online: 5 March 2018


Stern’s diatomic sequence with its intrinsic repetition and refinement structure between consecutive powers of $2$ gives rise to a rather natural probability measure on the unit interval. We construct this measure and show that it is purely singular continuous, with a strictly increasing, Hölder continuous distribution function. Moreover, we relate this function with the solution of the dilation equation for Stern’s diatomic sequence.


  • Michael BaakeFakultät für Mathematik
    Universität Bielefeld
    Postfach 100131
    33501 Bielefeld, Germany
  • Michael CoonsSchool of Mathematical and Physical Sciences
    University of Newcastle
    University Drive
    Callaghan, NSW 2308, Australia

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