AWhat changed
Successive convergents of a continued fraction have denominators q_n. These control both the size of the fractions and their approximation accuracy. Khintchine proves that the nth root of q_n approaches a common absolute constant for almost every number. His argument compares the full denominator ratio with a finite block and controls the truncation error. In a separate paper, Lévy identifies the constant as exp(pi squared divided by 12 log 2). Lévy explicitly credits Khintchine's existence result in a later-added note. This record preserves the existence and evaluation contributions as one result family, rather than inventing a joint paper.
Scientific scope
Successive convergents of a continued fraction have denominators q_n. These control both the size of the fractions and their approximation accuracy. Khintchine proves that the nth root of q_n approaches a common absolute constant for almost every number. His argument compares the full denominator ratio with a finite block and controls the truncation error. In a separate paper, Lévy identifies the constant as exp(pi squared divided by 12 log 2). Lévy explicitly credits Khintchine's existence result in a later-added note. This record preserves the existence and evaluation contributions as one result family, rather than inventing a joint paper. The claim is almost everywhere, not a bound for every irrational or a universal finite-term error guarantee. The constant describes denominator growth, not the geometric mean of the partial quotients. Khintchine's article was received in February 1935 and published in 1936; Lévy's added note records receipt in July 1935. Their earlier probability and continued-fraction inputs retain separate credit.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. The claim is almost everywhere, not a bound for every irrational or a universal finite-term error guarantee. The constant describes denominator growth, not the geometric mean of the partial quotients. Khintchine's article was received in February 1935 and published in 1936; Lévy's added note records receipt in July 1935. Their earlier probability and continued-fraction inputs retain separate credit.
Contributors
- The named authors in the research record · SCIENCE: Original-paper scientific authors for this exact mathematical contribution. Independent papers grouped in one historical family are not a joint byline. Earlier theorems, supporting proof contributions and later generalizations remain separate. Individual effort shares, author nationality and unresolved cross-record identities are not inferred.
- Zur metrischen Kettenbruchtheorie (Compositio Mathematica 3, 276–285)
- Sur le développement en fraction continue d’un nombre choisi au hasard (Compositio Mathematica 3, 286–303)
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