## Functiones et Approximatio Commentarii Mathematici

### Numbers with integer expansion in the numeration system with negative base

#### Abstract

In this paper, we study representations of real numbers in the positional numeration system with negative base, as introduced by Ito and Sadahiro. We focus on the set $\mathbb{Z}_{-\beta}$ of numbers whose representation uses only non-negative powers of $-\beta$, the so-called $(-\beta)$-integers. We describe the distances between consecutive elements of $\mathbb{Z}_{-\beta}$. In case that this set is non-trivial we associate to $\beta$ an infinite word $\boldsymbol{v}_{-\beta}$ over an (in general infinite) alphabet. The self-similarity of $\mathbb{Z}_{-\beta}$, i.e., the property $-\beta \\mathbb{Z}_{-\beta}\subset \mathbb{Z}_{-\beta}$, allows us to find a~morphism under which $\boldsymbol{v}_{-\beta}$ is invariant. On the example of two cubic irrational bases $\beta$ we demonstrate the difference between Rauzy fractals generated by $(-\beta)$-integers and by $\beta$-integers.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 47, Number 2 (2012), 241-266.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.facm/1356012918

Digital Object Identifier
doi:10.7169/facm/2012.47.2.8

Mathematical Reviews number (MathSciNet)
MR3051451

Zentralblatt MATH identifier
1271.11009

#### Citation

Ambrož, Petr; Dombek, Daniel; Masáková, Zuzana; Pelantová, Edita. Numbers with integer expansion in the numeration system with negative base. Funct. Approx. Comment. Math. 47 (2012), no. 2, 241--266. doi:10.7169/facm/2012.47.2.8. https://projecteuclid.org/euclid.facm/1356012918

#### References

• 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.
• P. Arnoux, S. Ito, Pisot substitutions and Rauzy fractals, Bull. Belg. Math. Soc. Simon Stevin 8 (2001), 181–207.
• 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.
• S. Fabre, Substitutions et $\beta$-systèmes de numération, Theoret. Comput. Sci. 137 (1995), 219–236.
• Ch. Frougny and A.C. Lai. On negative bases, In `Proceedings of DLT 09', Lectures Notes in Computer Science 5583 (2009), 252–263.
• Ch. Frougny and B. Solomyak, Finite $\beta$-expansions, Ergodic Theory Dynam. Systems 12 (1994), 713–723.
• S. Ito and T. Sadahiro, $(-\beta)$-expansions of real numbers, Integers 9 (2009), 239–259.
• C. Kalle, W. Steiner, Beta-expansions, natural extensions and multiple tilings associated with Pisot units, to appear in Trans. Amer. Math. Soc., (2010).
• Z. Masáková, E. Pelantová, T. Vávra, Arithmetics in number systems with a negative base, Theor. Comp. Sci. 412 (2011), 835–845.
• W. Parry, On the $\beta$-expansions of real numbers, Acta Math. Acad. Sci. Hung. 11 (1960), 401–416.
• A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hung. 8 (1957), 477–493.
• W. Steiner, On the structure of $(-\beta)$-integers, RAIRO – Theoretical Informatics and Applications 46 (2012), 181–200.
• W. Steiner, On the Delone property of $(-\beta)$-integers, in Proceedings WORDS 2011, EPTCS 63 (2011), 247–256.
• W.P. Thurston, Groups, tilings, and finite state automata, AMS Colloquium Lecture Notes, American Mathematical Society, Boulder, 1989.