GK-Regular

An irrational number  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  where  is a nonzero integer. Most widely-studied real numbers not of one of these forms, such as  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.