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Periodic representations in Salem bases

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Abstract

We prove that all algebraic bases β allow an eventually periodic representation of the elements of ℚ(β) with a finite alphabet of digits \({\cal A}\). Moreover, the classification of bases allowing that those representations have the socalled weak greedy property is given.

The decision problem whether a given pair (β, \({\cal A}\)) allows eventually periodic representations proves to be rather hard, for it is equivalent to a topological property of the attractor of an iterated function system.

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Acknowledgements

The author thanks the anonymous reviewer for many useful comments.

This work has been supported by Czech Science Foundation GAČR, grant 17-04703Y, and by Charles University Research Centre program No. UNCE/SCI/022.

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Authors and Affiliations

  1. Department of Algebra, Charles University, Sokolovská 83, 186 75, Praha 8, Czech Republic

    Tomáš Vávra

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  1. Tomáš Vávra
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Correspondence to Tomáš Vávra.

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Vávra, T. Periodic representations in Salem bases. Isr. J. Math. 242, 83–95 (2021). https://doi.org/10.1007/s11856-021-2123-3

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  • DOI: https://doi.org/10.1007/s11856-021-2123-3

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