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Periodic representations in Salem bases

Research paper by Tomáš Vávra

Indexed on: 19 Dec '18Published on: 19 Dec '18Published in: arXiv - Mathematics - Number Theory



Abstract

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.