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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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