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Difference between revisions of "Champernowne word"

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(Start article: Champernowne word)
 
 
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====References====
 
====References====
* Bugeaud, Yann. "Distribution modulo one and Diophantine approximation", Cambridge Tracts in Mathematics '''193'''' Cambridge University Press (2012) ISBN 978-0-521-11169-0 {{ZBL|1260.11001}}
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* Bugeaud, Yann. "Distribution modulo one and Diophantine approximation", Cambridge Tracts in Mathematics '''193'''' Cambridge University Press (2012) {{ISBN|978-0-521-11169-0}} {{ZBL|1260.11001}}
* Berthé, Valérie; Rigo, Michel. "Combinatorics, automata, and number theory". Encyclopedia of Mathematics and its Applications '''135''' Cambridge University Press ISBN 978-0-521-51597-9 {{ZBL|1216.68204}}
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* Berthé, Valérie; Rigo, Michel. "Combinatorics, automata, and number theory". Encyclopedia of Mathematics and its Applications '''135''' Cambridge University Press {{ISBN|978-0-521-51597-9}} {{ZBL|1216.68204}}

Latest revision as of 19:33, 6 December 2023

The infinite word over the alphabet $B = \{0,1,\ldots,b-1\}$ for given $b \ge 2$, obtained by concatenating the representations of the natural numbers with respect to base $b$. Thus for $b=2$ the word begins $$ 0\,1\,10\,11\,100\,101 \ldots \ . $$

This is a disjunctive word, that is, contains each finite sequence over $B$ as a factor infinitely often.

The Champernowne number is obtained by interpreting the word as the sequence of digits of a number $c_b$ in the interval $(0,1)$. The Champernowne number is a weakly normal number base $b$.

References

  • Bugeaud, Yann. "Distribution modulo one and Diophantine approximation", Cambridge Tracts in Mathematics 193' Cambridge University Press (2012) ISBN 978-0-521-11169-0 Zbl 1260.11001
  • Berthé, Valérie; Rigo, Michel. "Combinatorics, automata, and number theory". Encyclopedia of Mathematics and its Applications 135 Cambridge University Press ISBN 978-0-521-51597-9 Zbl 1216.68204
How to Cite This Entry:
Champernowne word. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Champernowne_word&oldid=39294