Mathematics > Number Theory
[Submitted on 19 Dec 2018]
Title:Periodic representations in Salem bases
View PDFAbstract:We prove that all algebraic bases $\beta$ allow an eventually periodic representations of the elements of $\mathbb Q(\beta)$ with a finite alphabet of digits $\mathcal A$. Moreover, the classification of bases allowing that those representations have the so-called weak greedy property is given.
The decision problem whether a given pair $(\beta,\mathcal A)$ 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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