Sturmian graphs and integer representations over numeration systems

Authors:
C. Epifanio;C. Frougny;A. Gabriele;F. Mignosi;J. Shallit
Affiliations:
Dipartimento di Matematica e Informatica, Universití di Palermo, Italy;LIAFA, CNRS & Universitéé Paris 7, and Université Paris 8, France;Dipartimento di Matematica e Informatica, Universití di Palermo, Italy;Dipartimento di Informatica, Universití degli Studi di L'Aquila, Italy;School of Computer Science, University of Waterloo, Ontario, Canada
Venue:
Discrete Applied Mathematics
Year:
2012

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Abstract

In this paper we consider a numeration system, originally due to Ostrowski, based on the continued fraction expansion of a real number @a. We prove that this system has deep connections with the Sturmian graph associated with @a. We provide several properties of the representations of the natural integers in this system. In particular, we prove that the set of lazy representations of the natural integers in this numeration system is regular if and only if the continued fraction expansion of @a is eventually periodic. The main result of the paper is that for any number i the unique path weighted i in the Sturmian graph associated with @a represents the lazy representation of i in the Ostrowski numeration system associated with @a.