Numbers with integer expansion in the numeration system with negative base. (English) Zbl 1271.11009

A. Rényi has introduced expansions of real numbers by positive base \(\beta> 1\) [Acta Math. Acad. Sci. Hung. 8, 477–493 (1957; Zbl 0079.08901)]. Next, S. Ito and T. Sadahiro [Integers 9, No. 3, 239–259, A22 (2009; Zbl 1191.11005)] generalized Rényi expansions for negative base \(-\beta< -1\).
In the present paper, the authors continue the study of Ito and Sadahiro [loc. cit.], providing intrinsic properties, as for instance the description of the distances between consecutive \((-\beta)\)-integers. Furthermore, it is introduced an infinite word \(u_{-\beta}\) over a finite alphabet, which is invariant under a primitive morphism.
Finally, in two examples of cubic Pisot numbers \(\beta\), the authors parallel the fractal tiles arising from \((-\beta)\)-expansions with the Rauzy fractal given by Rényi’s \(\beta\)-expansions.


11A63 Radix representation; digital problems
11K16 Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
37B10 Symbolic dynamics
68R15 Combinatorics on words
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