An irrational number x can be called GK-regular (defined here for the first time) if the distribution of its continued fraction coefficients is the Gauss-Kuzmin distribution. An irrational number that is not GK-regular is said to be GK-irregular. It has been shown that all real numbers except for a set of measure zero are GK-regular.

Classes of GK-irregular numbers include rational and real quadratic numbers and numbers of the form e^(2/n) where n is a nonzero integer. Most widely-studied real numbers not of one of these forms, such as pi and algebraic numbers of degree greater than 2, are strongly suspected to be GK-regular based on numerical evidence, although no proof is known. The situation is much the same for normal and absolutely normal numbers.

See also

Absolutely Normal, Gauss-Kuzmin Distribution, Normal Number

This entry contributed by David Terr

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Terr, David. "GK-Regular." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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