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(1935)
Compositio Math
G. Harman (1998)
Metric number theory
(1964)
Khinchin, Continued fractions (University of Chicago Press, Chicago, 1964)
(1935)
Khinchin, Metrische Kettenbruchprobleme
Álvaro Hernández (2019)
Continued fractionsQuadratic Number Theory
H. Davenport (1969)
On the Distribution of the Convergents of Almost All Real Numbers
A. Haynes, J. Vaaler (2008)
Martingale differences and the metric theory of continued fractionsIllinois Journal of Mathematics, 52
H. Diamond, J. Vaaler (1986)
ESTIMATES FOR PARTIAL SUMS OF CONTINUED FRACTION PARTIAL QUOTIENTSPacific Journal of Mathematics, 122
A. Yao, D. Knuth (1975)
Analysis of the subtractive algorithm for greatest common divisorsProceedings of the National Academy of Sciences of the United States of America, 72 12
A landmark theorem in the metric theory of continued fractions begins this way: Select a non‐negative real function f defined on the positive integers and a real number x, and form the partial sums sn of f evaluated at the partial quotients a1, …, an in the continued fraction expansion for x. Does the sequence {sn/n} have a limit as n → ∞? In 1935 Khinchin proved that the answer is yes for almost every x, provided that the function f does not grow too quickly. In this article we are going to explore a natural reformulation of this problem in which the function f is defined on the rationals and the partial sums in question are over the intermediate convergents to x with denominators less than a prescribed amount. By using some of Khinchin's ideas together with more modern results we are able to provide a quantitative asymptotic theorem analogous to the classical one mentioned above.
Bulletin of the London Mathematical Society – Wiley
Published: Jun 1, 2009
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