Sturmian graphs and integer representations over numeration systems

In this paper we consider a numeration system, originally due to Ostrowski, based on the continued fraction expansion of a real number alpha. We prove that this system has deep connections with the Sturmian graph associated with alpha. 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 alpha 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 alpha represents the lazy representation of i in the Ostrowski numeration system associated with alpha.