Champernowne constant

In mathematics, the Champernowne constant C₁₀ is a transcendental real constant whose decimal expansion has important properties. It is named after economist and mathematician D. G. Champernowne, who published it as an undergraduate in 1933.^[1]

For base 10, the number is defined by concatenating representations of successive integers:

C₁₀ = 0.12345678910111213141516…  (sequence A033307 in the OEIS).

Champernowne constants can also be constructed in other bases, similarly, for example:

C₂ = 0.11011100101110111… ₂

C₃ = 0.12101112202122… ₃.

The Champernowne constants can be expressed exactly as infinite series:

${\displaystyle C_{m}=\sum _{n=1}^{\infty }{\frac {n}{m^{~\left(\sum \limits _{k=1}^{n}\left\lceil \log _{m}(k+1)\right\rceil \right)}}}}$

where ${\displaystyle \lceil {x}\rceil =}$ ceiling(${\displaystyle x}$).^[2]

A slightly different expression (using floor instead of ceiling) is given by Eric W. Weisstein (MathWorld):

${\displaystyle C_{m}=\sum _{n=1}^{\infty }{\frac {n}{m^{\left(n+\sum \limits _{k=1}^{n-1}\left\lfloor \log _{m}(k+1)\right\rfloor \right)}}}}$

where ${\displaystyle \lfloor {x}\rfloor =}$ floor(${\displaystyle x}$).

Words and sequences

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.

Normality

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 in base ${\displaystyle b}$ for any b.^[6] It is an open problem whether ${\displaystyle C_{k}}$ is normal in bases ${\displaystyle b\neq k}$.

It is also a disjunctive sequence.

Continued fraction expansion

The first 161 quotients of the continued fraction of the Champernowne constant. The 4th, 18th, 40th, and 101st are much bigger than 270, so do not appear on the graph.

The first 161 quotients of the continued fraction of the Champernowne constant on a 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, ...]. (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 ${\displaystyle n}$. That is, such that the equation

${\displaystyle C_{10}=\sum _{q=1}^{\infty }{\biggl (}10^{-\delta (q)}\sum _{k=10^{q-1}}^{10^{q}-1}{\frac {k}{10^{q(k+1-10^{q-1})}}}{\biggr )}}$

, where ${\displaystyle \delta (q)={\frac {1}{9}}{\bigl (}1+10^{q-1}(9q-10){\bigr )}}$, becomes${\displaystyle C_{10}\cong \sum _{q=1}^{n-1}{\biggl (}10^{-\delta (q)}\sum _{k=10^{q-1}}^{10^{q}-1}{\frac {k}{10^{q(k+1-10^{q-1})}}}{\biggr )}+{\biggl (}10^{-\delta (n)}\sum _{k=10^{n-1}}^{\infty }{\frac {k}{10^{n(k+1-10^{n-1})}}}{\biggr )}}$

Where the second term can also be written as

${\displaystyle 10^{-\delta (n)}\sum _{k=10^{n-1}}^{\infty }{\frac {k}{10^{n(k+1-10^{n-1})}}}=10^{-\delta (n)}{\frac {10^{n-1}(10^{n}-1)+1}{(10^{n}-1)^{2}}}}$

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

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

which approximates Champernowne's constant with an error of about 1 × 10⁻⁹. While truncating just before the 18th partial quotient, using ${\displaystyle n=2}$, 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, ...

Irrationality measure

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]

See also

• Copeland–Erdős constant, a similar normal number, defined using the prime numbers
• Liouville's constant, another constant defined by its decimal representation
• Smarandache–Wellin number, another number obtained through concatenation a representation in a given base.

References

• Cassaigne, J.; Nicolas, F. (2010). "Factor complexity". In Berthé, Valérie; Rigo, Michel (eds.). Combinatorics, automata, and number theory. Encyclopedia of Mathematics and its Applications. Vol. 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.

External links

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