Champernowne constant

In mathematics, the Champernowne constant 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. The number is defined by concatenating the base-10 representations of the positive integers:
Champernowne constants can also be constructed in other bases similarly; for example,
and
The Champernowne word or Barbier word is the sequence of digits of C₁₀ obtained by writing it in base 10 and juxtaposing the digits:
More generally, a Champernowne sequence is any sequence of digits obtained by concatenating all finite digit-strings in some recursive order.
For instance, the binary Champernowne sequence in shortlex order is
where spaces have been inserted just to show the strings being concatenated.

Properties

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. A number 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, then, in base 10, we would expect strings,,, …, to occur 1/10 of the time, strings,,...,, to occur 1/100 of the time, and so on, in a normal number.
Champernowne proved that is normal in base 10, while Nakai and Shiokawa proved a more general theorem, a corollary of which is that is normal in base for any b. It is an open problem whether is normal in bases. For example, it is not known if is normal in base 9.
Kurt Mahler showed that the constant is transcendental.
The irrationality measure of is, and more generally for any base.
The Champernowne word is a disjunctive sequence. A disjunctive sequence is an infinite sequence in which every finite string appears as a substring.

Series

The definition of the Champernowne constant immediately gives rise to an infinite series representation involving a double sum,
where is the number of digits between the decimal point and the first contribution from an -digit base-10 number; these expressions generalize to an arbitrary base by replacing 10 and 9 with and respectively. Alternative forms are
and
where and denote the floor and ceiling functions.
Returning to the first of these series, both the summand of the outer sum and the expression for can be simplified using the closed form for the two-dimensional geometric series:
The resulting expression for is
while the summand of the outer sum becomes
Summing over all gives
Observe that in the summand, the expression in parentheses is approximately for and rapidly approaches that value as grows, while the exponent grows exponentially with. As a consequence, each additional term provides an exponentially growing number of correct digits even though the number of digits in the numerators and denominators of the fractions comprising these terms grows only linearly. For example, the first few terms of are

Continued fraction expansion

[Image:Champernowne constant.svg|right|177x177px|thumb|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.]
Image:Champernowne constant logscale.svg|right|172x172px|thumb|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 does not terminate and is aperiodic. A simple continued fraction is a continued fraction where the denominator is 1. The simple continued fraction expansion of Champernowne's constant exhibits extremely large terms appearing between many small ones. For example, in base 10,
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. That there are such large numbers as terms of the continued fraction expansion means that the convergents obtained by stopping before these large numbers provide an exceptionally good approximation of the Champernowne constant. For example, truncating just before the 4th partial quotient, gives
which matches the first term in the rapidly converging series expansion of the previous section and which approximates Champernowne's constant with an error of about. Truncating just before the 18th partial quotient gives an approximation that matches the first two terms of the series, that is, the terms up to the term containing,
which approximates Champernowne's constant with error approximately.
The first and second incrementally largest terms after the initial zero are 8 and 9, respectively, and occur at positions 1 and 2. Sikora noticed that the number of digits in the high-water marks starting with the fourth display an apparent pattern. Indeed, the high-water marks themselves grow doubly-exponentially, and the number of digits in the nth mark for are
whose pattern becomes obvious starting with the 6th high-water mark. The number of terms can be given by
However, it is still unknown as to whether or not there is a way to determine where the large terms occur, or their values. The high-water marks themselves are located at positions