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Multiple tilings associated to d-Bonacci beta-expansions

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Abstract

Let \(\beta \in (1,2)\) be a Pisot unit and consider the symmetric \(\beta \)-expansions. We give a necessary and sufficient condition for the associated Rauzy fractals to form a tiling of the contractive hyperplane. For \(\beta \) a d-Bonacci number, i.e., Pisot root of \(x^d-x^{d-1}-\dots -x-1\) we show that the Rauzy fractals form a multiple tiling with covering degree \(d-1\).

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Acknowledgements

We acknowledge support by Czech Science Foundation (GAČR) grant 17-04703Y and ANR/FWF project “FAN – Fractals and Numeration” (ANR-12-IS01-0002, FWF grant I1136). We are also grateful for Sage and TikZ software which were used to prepare the Figs. [13, 15].

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  1. IRIF, CNRS UMR 8243, Université Paris Diderot – Paris 7, Case 7014, 75205, Paris, France

    Tomáš Hejda

  2. Department of Mathematics FCE, University of Chemistry and Technology, Prague, Studentská 2031/6, Prague, 16000, Czechia

    Tomáš Hejda

  3. Department of Algebra FMF, Charles University, Sokolovská 49/83, Prague, 18600, Czechia

    Tomáš Hejda

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  1. Tomáš Hejda
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Correspondence to Tomáš Hejda.

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Communicated by H. Bruin.

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Hejda, T. Multiple tilings associated to d-Bonacci beta-expansions. Monatsh Math 187, 275–291 (2018). https://doi.org/10.1007/s00605-018-1219-2

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