SWhat changed
Fractions approximate irrational numbers, but how often can they meet a chosen accuracy target? Khintchine's 1924 theorem gives a threshold for almost every real number. In the original statement, a positive continuous function f specifies the error bound f(q)/q for denominator q, and x f(x) must decrease. If the improper integral of f diverges, almost every number has infinitely many fractions meeting that bound. If it converges, almost every number has only finitely many. Continued fractions connect the divergent case to earlier estimates of Bernstein. The contribution is this original metric threshold, rather than a claim about every particular irrational.
Scientific scope
Fractions approximate irrational numbers, but how often can they meet a chosen accuracy target? Khintchine's 1924 theorem gives a threshold for almost every real number. In the original statement, a positive continuous function f specifies the error bound f(q)/q for denominator q, and x f(x) must decrease. If the improper integral of f diverges, almost every number has infinitely many fractions meeting that bound. If it converges, almost every number has only finitely many. Continued fractions connect the divergent case to earlier estimates of Bernstein. The contribution is this original metric threshold, rather than a claim about every particular irrational. Almost every allows an exceptional set of Lebesgue measure zero, which can depend on f. The original continuity and monotonicity assumptions are retained; later weaker versions are not attributed to this paper. The manuscript is dated September 1923 and the volume was published in 1924. A preliminary geometric-mean bound in the same paper is distinct from the exact constant proved in 1935.
Evidence
The linked original works support the scoped statement and contribution attribution. WRS has not independently replayed experiments or reconstructed a full mathematical proof. Almost every allows an exceptional set of Lebesgue measure zero, which can depend on f. The original continuity and monotonicity assumptions are retained; later weaker versions are not attributed to this paper. The manuscript is dated September 1923 and the volume was published in 1924. A preliminary geometric-mean bound in the same paper is distinct from the exact constant proved in 1935.
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.
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