## Abstract $$\beta$$-expansions and ultimately periodic representations.(English)Zbl 1084.11059

A natural generalization of the radix expansion of a real number in integer base $$b \geq 2$$ is a so-called $$\beta$$-expansion introduced by Rényi in 1957, where $$\beta>1$$ is an arbitrary real number. It is known that the set of real numbers with ultimately periodic $$\beta$$-expansions is $${\mathbb Q}(\beta)$$ whenever $$\beta$$ is a Pisot number. In this paper, the authors study infinite regular languages $$L$$ over a finite and totally ordered alphabet $$(\Sigma, <).$$ In terms of certain ordering of words the set $$L$$ can be enumerated in some order leading to a one-to-one correspondence between $$L$$ and the set of positive integers $${\mathbb N}.$$ The triplet $$(L,\Sigma,<)$$ is called an abstract numeration system. If $$w$$ is the $$n$$th word of $$L$$, then $$n$$ is called the numerical value of $$w.$$ Equivalently, $$w$$ is said to be the representation of $$n.$$ In this way, under some natural restrictions, the authors introduce the representations of real numbers via those abstract numeration systems. Generally speaking, the representation of a number is not necessarily unique. The role of $$\beta$$ is played by the dominant eigenvalue of the automaton accepting the language. Generalizing previous results, the authors show that the set of real numbers having ultimately periodic expansions is $${\mathbb Q}(\beta)$$ if $$\beta$$ is a Pisot number. Moreover, if $$\beta$$ is neither Pisot nor Salem number then there exist points in $${\mathbb Q}(\beta)$$ such that none of their expansions is ultimately periodic. They conclude with few examples showing how their theory works in practice.

### MSC:

 11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure 03D45 Theory of numerations, effectively presented structures
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### References:

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