Champernowne word

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