Champernowne constant

The Champernowne word or Barbier word is the sequence of digits of C₁₀, obtained writing n in base 10 and juxtaposing the digits:[3][4]

12345678910111213141516…  (sequence A007376 in the OEIS)

More generally, a Champernowne sequence (sometimes also called a Champernowne word) is any sequence of digits obtained by concatenating all finite digit-strings (in any given base) in some recursive order.[5] For instance, the binary Champernowne sequence in shortlex order is

0 1 00 01 10 11 000 001 ... (sequence A076478 in the OEIS)

where spaces (otherwise to be ignored) have been inserted just to show the strings being concatenated.

A real number x is said to be normal if its digits in every base follow a uniform distribution: all digits being equally likely, all pairs of digits equally likely, all triplets of digits equally likely, etc. x is said to be normal in base b if its digits in base b follow a uniform distribution.

If we denote a digit string as [a₀,a₁,...], then, in base 10, we would expect strings [0],[1],[2],...,[9] to occur 1/10 of the time, strings [0,0],[0,1],...,[9,8],[9,9] to occur 1/100 of the time, and so on, in a normal number.

Champernowne proved that ${\displaystyle C_{10}}$ is normal in base 10,[1] while Nakai and Shiokawa proved a more general theorem, a corollary of which is that ${\displaystyle C_{b}}$ is normal for any base ${\displaystyle b}$.[6] It is an open problem whether ${\displaystyle C_{k}}$ is normal in bases ${\displaystyle b\neq k}$.

It is also disjunctive sequence.

The first 161 quotients of the continued fraction of the Champernowne constant. The 4th, 18th, 40th, and 101st are (much) bigger than 270, so they don't appear in the graph.

The first 161 quotients of the continued fraction of the Champernowne constant represented using the logarithmic scale.

The simple continued fraction expansion of Champernowne's constant has been studied as well. Kurt Mahler showed that the constant is transcendental;[7] therefore its continued fraction does not terminate (because it is not rational) and is aperiodic (because it is not an irreducible quadratic).

The terms in the continued fraction expansion exhibit very erratic behaviour, with extremely large terms appearing between many small ones. For example, in base 10,

C₁₀ = [0; 8, 9, 1, 149083, 1, 1, 1, 4, 1, 1, 1, 3, 4, 1, 1, 1, 15, 4 57540 11139 10310 76483 64662 82429 56118 59960 39397 10457 55500 06620 04393 09026 26592 56314 93795 32077 47128 65631 38641 20937 55035 52094 60718 30899 84575 80146 98631 48833 59214 17830 10987, 6, 1, 1, 21, 1, 9, 1, 1, 2, 3, 1, 7, 2, 1, 83, 1, 156, 4, 58, 8, 54, 4457 35380 09111 78833 95906 76716 34293 78843 72929 58096 32494 71885 56700 06787 76593 24583 93083 78747 99958 33334 44419 14423 03860 34915 17196 39390 48693 60248 63329 21365 55338 06902 11515 02125 41996 97493 09604 01502 95147 16863 48283 22062 25872 13786 21729 02280 18637 91830 20719 74781 09902 20775 54858 29075 81744 18240 79663 01047 98318 61401 36632 17588 22094 34133 50221 71045 00117 00191 77903 31666 86409 06738 86569 58593 04299 72943 26256 16050 61474 79404 41223 75814 31178 71042 85903 73244 25563 98834 52924 26845 57032 80115 35178 53058 65904 33904 34647 30117 77033 64139 10737 82729 29235 36707 94698 74396 96805 36706 64437 92323 12459 93262 76905 56208 03487 13657 07092 74219 81334 81168 43029 92142 66535 78828 94275 64686 01860 23055 81293 42260 13213 60811 94088 52956 75626 03365 35724 56887 11887 04411 49430 03894 97130 17554 76895 68499 85837 12518 37087 09628 37893 82726 55423 11115 93763 30765 97881 76079 49380 33878 02379 39386 06324 84649 52301 68051 14768 53411 40243 43278 06603 52030 42618 93932 84476 36238 85961 73869 08570 88770 61190 14828 79211 45363 66498 29183 29626 75071 63929 06656 70416 26636 77687 18462 68218 08752 44878 33080 26657 99866 31150 86649 49675 12306 25639 37996 07172 04151 76470 27922 91223 87257 03496 02789 35019 74974 26361 33868 11757 07771 70606 71595 18277 90505 87307 14524 67326 49030 28322 32660 71730 63323 25765 52268 69072 64385 99565 21273 44604 98609 66734 11714 23675 65221 05392 62210 68825 60332 38380 98315 14785 92895 79327 59264 78437 49625 64499 98457 57920 77811 17862 02721 93172 63561 40910 94582 06988 25489 64745 29861 41136 55575 69779 67033 66788 69838 57950 62663 74602 91223 62030 75776 32910 70756 54119 76757 31110 01633 00236 51833 57876 63752 29235 90055 48386 59012 03305 69890 50164 06278 72889 11399 45000 48124 05460 03027 10228 35335 42123 83946 63495 47967 80419 61436 18818 99447 83761 54165 45986 17889 21370 92796 37606 41697 79765 24200 94822 80347 96282 12448 07192 83048 52209 75757 66277 46716 08588 92591 11902 85956 93963 81651 06653 33381 82474 78017 02273 19165 73419 80314 41320 89929 80545 59923 80497 47128 33350 15768 59373 77601 80648 66701 32693 42789 14682 29303 12242 33219 49257 15884 27105 60927 72035 82180 93087 33903 38309 60421 26830 89654 44227 28143 31276 56471 10757 19179 04603 69388 75682 38634 03667 52095 47642 08315 86673 60563 79327 65093 67564 83460 22985 33145 31244 46479 82468 87924 98984 85262 57924 81955 34239 83504 68427 19106 47612 92175 40957 38104 80927 62214 45354 37826 20910 50691 91508 66393 89789 84357 63014 34160 14809 37973 37580 34340 31052 62207 80365 93783 65162 60303 15374 71481 07560 84867 35866 18984 29025 52830 77170 68656 09828 61156 27168 05111 17630 19267 79997 44134 29491 26377 86664 15613 44560 94667 15529 33342 71641 59224 18499 89904 32427 72648 51963 63982 74050 10882 08925 32195 87625 98015 76960 06297 51352 40951 14112 26199 30775 83545 27328 15697 82833 35126 33155 42472 19337 67593 64768 62326 01763 51131 72423, 2, 3, 1, 3, 1, 2, 14, 1, 1, 1, 1, 1, 2, 3, 1, 2, 155, 1, 1, 1, 1, 7, 6, 1, 4, 8, 4, 2, 1, 11, 1, 1, 1, 8, 2, 30, 1, 3, 6, 2, 6, 7, 1, 2, 3, 2, 1, 2, 7, 1, 2, 5, 2, 6, 1, 4, 2, 1, 7, 1, 97867 20807 94824 38309 76939 66787 09858 47251 56169 85153 70452 33767 85637 40648 94306 92260 50302 98273 84487 61589 38858 70400 89991 59643 01331 83929 69448 67434, 7, 72, 1, 1, 13, 1, 4, 1, 6, 2, 1, 268, 1, 1, 40, 1, 5, 1, 6, 1, 1, 4, 1, 12, 3, 1, 4, 2, 1, 1, 18, 7, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 5, 1, 1, 13, 3, 73, 3, 1, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, ...]. (sequence A030167 in the OEIS)

The large number at position 18 has 166 digits, and the next very large term at position 40 of the continued fraction has 2504 digits. The fact that there are such large numbers as terms of the continued fraction expansion is equivalent to saying that the convergents obtained by stopping before these large numbers provide an exceptionally good approximation of the Champernowne constant.

It can be understood from infinite series expression of ${\displaystyle C_{10}}$: for a specified ${\displaystyle n}$ we can always approximate the sum over ${\displaystyle k}$ by setting the upper limit to ${\displaystyle \infty }$ instead of ${\displaystyle 10^{n}-1}$. Then we ignore the terms for higher ${\displaystyle n}$.

For example, if we keep lowest order of n, it is equivalent to truncating before the 4th partial quotient, we obtain the partial sum

${\displaystyle 10/81=\sum _{n=1}^{\infty }n/10^{n}=0.{\overline {123456790}}}$

which approximates Champernowne's constant with an error of about 1 × 10⁻⁹. While truncating just before the 18th partial quotient, we get the approximation to second order:

${\displaystyle {\begin{aligned}{\frac {60499999499}{490050000000}}&=0.123456789+10^{-9}\sum _{k=10}^{\infty }k/10^{2(k-9)}=0.123456789+10^{-9}{\frac {991}{9801}}\\&=0.123456789{\overline {10111213141516171819\ldots 90919293949596979900010203040506070809}},\end{aligned}}}$

which approximates Champernowne's constant with error approximately 9 × 10⁻¹⁹⁰.

The first and second incrementally largest terms ("high-water marks") after the initial zero are 8 and 9, respectively, and occur at positions 1 and 2. Sikora (2012) noticed that the number of digits in the high-water marks starting with the fourth display an apparent pattern[8]. Indeed, the high-water marks themselves grow doubly-exponentially, and the number of digits ${\displaystyle d_{n}}$ in the nth mark for ${\displaystyle n\geqslant 3}$ are:

6, 166, 2504, 33102, 411100, 4911098, 57111096, 651111094, 7311111092,...

whose pattern becomes obvious starting with the 6th high-water mark. The number of terms can be given by:

${\displaystyle d_{n}={\frac {13-67*10^{n-3}}{45}}+\left(2^{n}5^{n-3}-2\right),n\in \mathbb {Z} \cap \left[3,\infty \right)}$

However, it is still unknown as to whether or not there is a way to determine where the large terms (with at least 6 digits) occur, or their values. The high-water marks themselves, however, are located at positions:

1, 2, 4, 18, 40, 162, 526, 1708, 4838, 13522, 34062, ...

The irrationality measure of ${\displaystyle C_{10}}$ is ${\displaystyle \mu (C_{10})=10}$, and more generally ${\displaystyle \mu (C_{b})=b}$ for any base ${\displaystyle b\geq 2}$.[9]

• Copeland–Erdős constant, a similar normal number, defined using the prime numbers
• Liouville's constant, another constant defined by its decimal representation

• Cassaigne, J.; Nicolas, F. (2010). "Factor complexity". In Berthé, Valérie; Rigo, Michel (eds.). Combinatorics, automata, and number theory. Encyclopedia of Mathematics and its Applications. 135. Cambridge: Cambridge University Press. pp. 163–247. ISBN 978-0-521-51597-9. Zbl 1216.68204.
• Champernowne, D. G. (1933), "The construction of decimals normal in the scale of ten", Journal of the London Mathematical Society, 8 (4): 254–260, doi:10.1112/jlms/s1-8.4.254.
• Nakai, Y.; Shiokawa, I. (1992), "Discrepancy estimates for a class of normal numbers", Acta Arithmetica, 62 (3): 271–284, doi:10.4064/aa-62-3-271-284.

• Weisstein, Eric W. "Champernowne constant". MathWorld.