= Champernowne constant
{c}
{title2=$C_{10}=0.123456789101112\ldots$}
{wiki}
Form <Champernowne constant> by using <concatenation> to join the <decimal representations> of the <positive integers> $1,2,3,\ldots$ after the decimal point. Champernowne proved in 1933 that this particular <real number> is a <normal number> in base ten. The construction is striking because its rule is completely explicit, while all finite <words> have the prescribed limiting <digit block frequencies>. https://doi.org/10.1112/jlms/s1-8.4.254[D. G. Champernowne, The Construction of Decimals Normal in the Scale of Ten, Journal of the London Mathematical Society 8, 254–260, 1933]{ref} Copeland and Erdős explicitly recall this result at the start of their own paper. https://users.renyi.hu/~p_erdos/1946-01.pdf[Copeland and Erdős, Note on Normal Numbers, 1946, first paragraph]{ref}
The first $100000$ digits of <Champernowne constant> have the following exact counts, in digit order $0,1,\ldots,9$:
$$
(8642,19753,11111,8642,8642,8642,8642,8642,8642,8642).
$$
For the $99999$ adjacent pairs fully inside that prefix, the <words> $00,01,12,99$ occur $642,1753,2088,753$ times respectively. Thus, for example, $F_{100000}(12)=2088/99999$, which is about $0.02088$, rather than $0.01$. The prefix ends partway through the <decimal representation> of $22222$; the many leading ones in earlier <positive integers> help explain the visible imbalance.
\Image[/digit-block-frequency-comparison.png]
{title=Exact digit and adjacent-pair frequencies in two prefixes of 100000 decimal digits}
{description=The upper panels compare single-digit frequencies and list exact counts. The lower panels show all hundred overlapping pair frequencies, each with denominator $99999$, on the same vertical scale. The dashed $1/100$ line is the limiting reference for a <normal number> in base ten. In the periodic prefix, pairs $01$ through $89$ occur $10000$ times each and $90$ occurs $9999$ times; all other pairs are absent.}
These are finite counts, not a proof of the <normal number> theorem: a finite prefix can be followed by an all-zero tail. The theorem for <Champernowne constant> concerns the infinite <limit>, and establishes base-ten normality, rather than <absolute normality>. Nor do computations prove normality for familiar constants: the normality of <Pi> and <Euler's number> remains unproved, as discussed by Bailey and Borwein. https://www.davidhbailey.com/dhbpapers/normal-pseudo.pdf[David H. Bailey and Jonathan M. Borwein, Normal Numbers and Pseudorandom Generators, author manuscript, 2011, introduction]{ref}
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