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Is there any irrational number that is known the probability distribution of digits?

Something like 0 appears 10% of time, 1 appears 10% of time, etc.

Probably irrational numbers that are defined by a construction on digits like:

1234567891011121314....

You can prove digits distribution but I am interested in numbers that is not defined this way like pi, e, or square root of prime.

Is there any advance on this topic?

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    $\begingroup$ To what extent does en.wikipedia.org/wiki/Normal_number#Properties_and_examples not answer your question? The bottom line seems to be that normal numbers (which are by necessity irrational) are known, but none of the classical (computable) irrational numbers like $\pi$ are known to be normal. It is a notorious problem, and it seems to get asked here frequently. Do a search on "normal number" here in MO. $\endgroup$ – Todd Trimble Nov 5 '17 at 17:57
  • $\begingroup$ I expect that for every probability distribution, one can construct an irrational whose digits have that distribution. $\endgroup$ – Gerry Myerson Nov 5 '17 at 21:36
  • $\begingroup$ In brief and informally: the only way known to know that you have a specific distribution on the base $10$ digits of $r$ is to define $r$ via its base $10$ digits (perhaps adding a rational, which can always be defined as an eventually repeating decimal). There is every reason to think that anything else one can describe is normal. The normal reals constitute $99.9999+ \%$ of the reals (all but measure $0$) but no proofs for specific reals to be normal or not are known. $\endgroup$ – Aaron Meyerowitz Nov 6 '17 at 20:45
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If all you care about is the distribution of digits, it is easy to construct a number with a distribution arbitrarily close to a given one: just construct a rational number with a repeating block with close to the right distribution, then add to it a Liouville number (where the digits are zero, except for the ones in the $n!$ places. What is more, the above construction will work on the nose when the probabilities of digits are rational, and any number where the probability of some digit is irrational has to be irrational itself (exercise to the reader).

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Your question belongs to the context of normal numbers, which are real numbers whose digits and blocks of digits are uniformly distributed in the sense that each digits (or block of digits) occurs asymptotically with the correct "fair" frequency. The number you suggest is called "Champernowne's constant", it is known to be normal in base 10.

For a first introduction, see for example here: https://www.claymath.org/library/annual_report/ar2006/06report_normalnumbers.pdf

A number which is normal in all possible (integer) bases is called absolutely normal.

If the distribution of digits is not "fair", then the number is sometimes called "abnormal". See for example here: Martin, Greg. Absolutely abnormal numbers. Amer. Math. Monthly 108 (2001), no. 8, 746–754.

It is not known whether $\pi$, $e$, $\sqrt{2}$ etc. are normal in any base. However, it is conjectured that all algebraic irrationals are absolutely normal. See for example here for more information: Bailey, David H.; Crandall, Richard E.: On the random character of fundamental constant expansions. Experiment. Math. 10 (2001), no. 2, 175–190.

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