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.


11R06 PV-numbers and generalizations; other special algebraic numbers; Mahler measure
03D45 Theory of numerations, effectively presented structures
Full Text: DOI Numdam EuDML


[1] A. Bertrand, Développements en base de Pisot et répartition modulo \(1\). C. R. Acad. Sc. Paris 285 (1977), 419-421. · Zbl 0362.10040
[2] V. Bruyère, G. Hansel, Bertrand numeration systems and recognizability. Latin American Theoretical INformatics (Valparaíso, 1995). Theoret. Comput. Sci. 181 (1997), 17-43. · Zbl 0957.11015
[3] J.-M. Dumont, A. Thomas, Systèmes de numération et fonctions fractales relatifs aux substitutions. J. Theoret. Comput. Sci. 65 (1989), 153-169. · Zbl 0679.10010
[4] S. Eilenberg, Automata, languages, and machines. Vol. A, Pure and Applied Mathematics, Vol. 58, Academic Press , New York (1974). · Zbl 0317.94045
[5] C. Frougny, B. Solomyak, On representation of integers in linear numeration systems. In Ergodic theory of \(Z_d\) actions (Warwick, 1993-1994), 345-368, London Math. Soc. Lecture Note Ser. 228, Cambridge Univ. Press, Cambridge (1996). · Zbl 0856.11007
[6] C. Frougny, Numeration systems. In M. Lothaire, Algebraic combinatorics on words, Encyclopedia of Mathematics and its Applications 90. Cambridge University Press, Cambridge (2002).
[7] P. B. A. Lecomte, M. Rigo, Numeration systems on a regular language. Theory Comput. Syst. 34 (2001), 27-44. · Zbl 0969.68095
[8] P. Lecomte, M. Rigo, On the representation of real numbers using regular languages. Theory Comput. Syst. 35 (2002), 13-38. · Zbl 0993.68050
[9] P. Lecomte, M. Rigo, Real numbers having ultimately periodic representations in abstract numeration systems. Inform. and Comput. 192 (2004), 57-83. · Zbl 1055.11005
[10] W. Parry, On the \(β \)-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960), 401-416. · Zbl 0099.28103
[11] A. Rényi, Representation for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957), 477-493. · Zbl 0079.08901
[12] K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12 (1980), 269-278. · Zbl 0494.10040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.