Rational base number systems for p-adic numbers

Christiane Frougny; Karel Klouda

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications (2012)

  • Volume: 46, Issue: 1, page 87-106
  • ISSN: 0988-3754

Abstract

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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.

How to cite

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Frougny, Christiane, and Klouda, Karel. "Rational base number systems for p-adic numbers." RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications 46.1 (2012): 87-106. <http://eudml.org/doc/272986>.

@article{Frougny2012,
abstract = {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.},
author = {Frougny, Christiane, Klouda, Karel},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications},
keywords = {rational base number systems; p-adic numbers; -adic numbers},
language = {eng},
number = {1},
pages = {87-106},
publisher = {EDP-Sciences},
title = {Rational base number systems for p-adic numbers},
url = {http://eudml.org/doc/272986},
volume = {46},
year = {2012},
}

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
PB - EDP-Sciences
VL - 46
IS - 1
SP - 87
EP - 106
AB - 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.
LA - eng
KW - rational base number systems; p-adic numbers; -adic numbers
UR - http://eudml.org/doc/272986
ER -

References

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  1. [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. Zbl1214.11089MR2448050
  2. [2] I. Kátai and J. Szabó, Canonical number systems for complex integers. Acta Sci. Math. (Szeged) 37 (1975) 255–260. Zbl0309.12001MR389759
  3. [3] M. Lothaire, Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications 95. Cambridge University Press (2002). Zbl1221.68183MR1905123
  4. [4] K. Mahler, An unsolved problem on the powers of 3/2. J. Austral. Math. Soc.8 (1968) 313–321. Zbl0155.09501MR227109
  5. [5] M.R. Murty, Introduction to p-adic analytic number theory. American Mathematical Society (2002). Zbl1031.11067MR1913413
  6. [6] A. Odlyzko and H. Wilf, Functional iteration and the Josephus problem. Glasg. Math. J.33 (1991) 235–240. Zbl0751.05007MR1108748
  7. [7] A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar.8 (1957) 477–493. Zbl0079.08901MR97374
  8. [8] W.J. Robinson, The Josephus problem. Math. Gaz.44 (1960) 47–52. MR117163
  9. [9] J. Sakarovitch, Elements of Automata Theory. Cambridge University Press, New York (2009). Zbl1188.68177MR2567276

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