## 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.

### MSC:

 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

### Keywords:

representations of real numbers; Pisot numbers

### Citations:

Zbl 0079.08901; Zbl 1191.11005
Full Text:

### References:

 [1] S. Akiyama, Pisot number system and its dual tiling , In “,Physics and Theoretical Computer Science”, ed. by J.P. Gazeau et al., IOS Press (2007), 133-154. [2] P. Arnoux, S. Ito, Pisot substitutions and Rauzy fractals , Bull. Belg. Math. Soc. Simon Stevin 8 (2001), 181-207. · Zbl 1007.37001 [3] Č. Burdí k, Ch. Frougny, J.P. Gazeau, R. Krejcar, Beta-Integers as Natural Counting Systems for Quasicrystals , J. Phys. A: Math. Gen. 31 (1998), 6449-6472. · Zbl 0941.52019 [4] S. Fabre, Substitutions et $$\beta$$-systèmes de numération , Theoret. Comput. Sci. 137 (1995), 219-236. · Zbl 0872.11017 [5] Ch. Frougny and A.C. Lai. On negative bases , In ‘Proceedings of DLT 09’, Lectures Notes in Computer Science 5583 (2009), 252-263. · Zbl 1247.68139 [6] Ch. Frougny and B. Solomyak, Finite $$\beta$$-expansions , Ergodic Theory Dynam. Systems 12 (1994), 713-723. · Zbl 0814.68065 [7] S. Ito and T. Sadahiro, $$(-\beta)$$-expansions of real numbers , Integers 9 (2009), 239-259. · Zbl 1191.11005 [8] C. Kalle, W. Steiner, Beta-expansions, natural extensions and multiple tilings associated with Pisot units , to appear in Trans. Amer. Math. Soc., (2010). · Zbl 1295.11010 [9] Z. Masáková, E. Pelantová, T. Vávra, Arithmetics in number systems with a negative base , Theor. Comp. Sci. 412 (2011), 835-845. · Zbl 1226.11015 [10] W. Parry, On the $$\beta$$-expansions of real numbers , Acta Math. Acad. Sci. Hung. 11 (1960), 401-416. · Zbl 0099.28103 [11] A. Rényi, Representations for real numbers and their ergodic properties , Acta Math. Acad. Sci. Hung. 8 (1957), 477-493. · Zbl 0079.08901 [12] W. Steiner, On the structure of $$(-\beta)$$-integers , RAIRO - Theoretical Informatics and Applications 46 (2012), 181-200. · Zbl 1319.11006 [13] W. Steiner, On the Delone property of $$(-\beta)$$-integers , in Proceedings WORDS 2011, EPTCS 63 (2011), 247-256. · Zbl 1331.11004 [14] W.P. Thurston, Groups, tilings, and finite state automata , AMS Colloquium Lecture Notes, American Mathematical Society, Boulder, 1989.
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.