Rational base number systems for $p$-adic numbers
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 46 (2012) no. 1, pp. 87-106.

This paper deals with rational base number systems for p-adic numbers. We mainly focus on the system proposed by Akiyama et al. in 2008, but we also show that this system is in some sense isomorphic to some other rational base number systems by means of finite transducers. We identify the numbers with finite and eventually periodic representations and we also determine the number of representations of a given p-adic number.

DOI : https://doi.org/10.1051/ita/2011114
Classification : 11A67,  11E95
Mots clés : rational base number systems, p-adic numbers
@article{ITA_2012__46_1_87_0,
author = {Frougny, Christiane and Klouda, Karel},
title = {Rational base number systems for $p$-adic numbers},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
pages = {87--106},
publisher = {EDP-Sciences},
volume = {46},
number = {1},
year = {2012},
doi = {10.1051/ita/2011114},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ita/2011114/}
}
TY  - JOUR
AU  - Frougny, Christiane
AU  - Klouda, Karel
TI  - Rational base number systems for $p$-adic numbers
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2012
DA  - 2012///
SP  - 87
EP  - 106
VL  - 46
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ita/2011114/
UR  - https://doi.org/10.1051/ita/2011114
DO  - 10.1051/ita/2011114
LA  - en
ID  - ITA_2012__46_1_87_0
ER  - 
Frougny, Christiane; Klouda, Karel. Rational base number systems for $p$-adic numbers. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 46 (2012) no. 1, pp. 87-106. doi : 10.1051/ita/2011114. http://www.numdam.org/articles/10.1051/ita/2011114/

[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. | MR 2448050 | Zbl 1214.11089

[2] I. Kátai and J. Szabó, Canonical number systems for complex integers. Acta Sci. Math. (Szeged) 37 (1975) 255-260. | MR 389759 | Zbl 0309.12001

[3] M. Lothaire, Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications 95. Cambridge University Press (2002). | MR 1905123 | Zbl 1221.68183

[4] K. Mahler, An unsolved problem on the powers of 3/2. J. Austral. Math. Soc. 8 (1968) 313-321. | MR 227109 | Zbl 0155.09501

[5] M.R. Murty, Introduction to p-adic analytic number theory. American Mathematical Society (2002). | MR 1913413 | Zbl 1031.11067

[6] A. Odlyzko and H. Wilf, Functional iteration and the Josephus problem. Glasg. Math. J. 33 (1991) 235-240. | MR 1108748 | Zbl 0751.05007

[7] A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957) 477-493. | MR 97374 | Zbl 0079.08901

[8] W.J. Robinson, The Josephus problem. Math. Gaz. 44 (1960) 47-52. | MR 117163

[9] J. Sakarovitch, Elements of Automata Theory. Cambridge University Press, New York (2009). | MR 2567276 | Zbl 1188.68177

Cité par Sources :