Abstract
We study expansions in non-integer negative base − β introduced by Ito and Sadahiro [7]. Using countable automata associated with ( − β)-expansions, we characterize the case where the ( − β)-shift is a system of finite type. We prove that, if β is a Pisot number, then the ( − β)-shift is a sofic system. In that case, addition (and more generally normalization on any alphabet) is realizable by a finite transducer.
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References
Bertrand, A.: Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sci. Paris Sér A-B 285, A419–A421 (1977)
Eilenberg, S.: Automata, Languages and Machines, vol. A. Academic Press, London (1974)
Fotiades, N., Boudourides, M.: Topological conjugacies of piecewise monotone interval maps. Int. J. Math. Math. Sci. 25, 119–127 (2001)
Frougny, C.: Representations of numbers and finite automata. Math. Systems Theory 25, 37–60 (1992)
Frougny, C.: On-line finite automata for addition in some numeration systems. Theoretical Informatics and Applications 33, 79–101 (1999)
Grünwald, V.: Intorno all’aritmetica dei sistemi numerici a base negativa con particolare riguardo al sistema numerico a base negativo-decimale per lo studio delle sue analogie coll’aritmetica ordinaria (decimale). Giornale di Matematiche di Battaglini 367, 203–221 (1885)
Ito, S., Sadahiro, T.: ( − β)-expansions of real numbers. In: INTEGERS (to appear)
Knuth, D.E.: The Art of Computer Programming, Seminumerical Algorithms, 2nd edn., vol. 2. Addison-Wesley, Reading (1988)
Lind, D., Marcus, B.: An introduction to symbolic dynamics and coding. Cambridge University Press, Cambridge (1995)
Lothaire, M.: Algebraic combinatorics on words. Encyclopedia of Mathematics and its Applications, vol. 90. Cambridge University Press, Cambridge (2002)
Parry, W.: On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11, 401–416 (1960)
Rényi, A.: Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8, 477–493 (1957)
Sakarovitch, J.: Eléments de théorie des automates, Vuibert (2003); English translation: Elements of Automata Theory, Cambridge University Press, Cambridge (to appear)
Schmidt, K.: On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12, 269–278 (1980)
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Frougny, C., Lai, A.C. (2009). On Negative Bases. In: Diekert, V., Nowotka, D. (eds) Developments in Language Theory. DLT 2009. Lecture Notes in Computer Science, vol 5583. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02737-6_20
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DOI: https://doi.org/10.1007/978-3-642-02737-6_20
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