×

Rational base number systems for \(p\)-adic numbers. (English) Zbl 1338.11010

This paper is concerned with rational base number systems for \(p\)-adic numbers, in particular with those introduced in [S. Akiyama et al., Isr. J. Math. 168, 53–91 (2008; Zbl 1214.11089)]. The authors identify those numbers which have finite and eventually periodic representations, and they determine the number of representations of a given \(p\)-adic number. Moreover, the authors investigate the relation between this number system and three other rational base number systems for \(p\)-adic numbers, and show that they all are in some sense isomorphic and share most of their properties.

MSC:

11A67 Other number representations
11S99 Algebraic number theory: local fields

Citations:

Zbl 1214.11089
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] S. Akiyama, Ch. Frougny and J. Sakarovitch, Powers of rationals modulo 1 and rational base number systems. Isr. J. Math.168 (2008) 53-91. · Zbl 1214.11089 · doi:10.1007/s11856-008-1056-4
[2] I. Kátai and J. Szabó, Canonical number systems for complex integers. Acta Sci. Math. (Szeged)37 (1975) 255-260. · Zbl 0309.12001
[3] M. Lothaire, Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications95. Cambridge University Press (2002). · Zbl 1001.68093
[4] K. Mahler, An unsolved problem on the powers of 3/2. J. Austral. Math. Soc.8 (1968) 313-321. · Zbl 0155.09501
[5] M.R. Murty, Introduction to p-adic analytic number theory. American Mathematical Society (2002). · Zbl 1031.11067
[6] A. Odlyzko and H. Wilf, Functional iteration and the Josephus problem. Glasg. Math. J.33 (1991) 235-240. · Zbl 0751.05007 · doi:10.1017/S0017089500008272
[7] A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar.8 (1957) 477-493. · Zbl 0079.08901 · doi:10.1007/BF02020331
[8] W.J. Robinson, The Josephus problem. Math. Gaz.44 (1960) 47-52.
[9] J. Sakarovitch, Elements of Automata Theory. Cambridge University Press, New York (2009). · Zbl 1188.68177
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.