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Sturmian graphs and integer representations over numeration systems. (English) Zbl 1237.68134
Summary: 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 $\alpha$.

##### MSC:
 11A67 Representation systems for integers and rationals 05C75 Structural characterization of families of graphs 05C90 Applications of graph theory 68R15 Combinatorics on words
##### Keywords:
numeration systems; Sturmian graphs; continued fractions