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Proceedings of the American Mathematical Society

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The diophantine approximation of certain continued fractions


Author: Charles F. Osgood
Journal: Proc. Amer. Math. Soc. 36 (1972), 1-7
MSC: Primary 10F05
DOI: https://doi.org/10.1090/S0002-9939-1972-0308059-X
MathSciNet review: 0308059
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Abstract | References | Similar Articles | Additional Information

Abstract: Given a real number $ \alpha $ defined by

$\displaystyle \frac{1}{{\varphi (1) + }}\frac{1}{{\varphi (2) + }} \cdots ,$

where $ \varphi $ is a function from the natural numbers to the rational numbers larger than or equal to one which satisfies certain restrictions on the growth of the numerators and denominators of the numbers $ \varphi (n)$, then a lower bound is found in terms of $ \varphi $ for the diophantine approximation of $ \alpha $.

References [Enhancements On Off] (What's this?)

  • [1] A. Ja. Hinčin, Continued fractions, GITTL, Moscow, 1961 ; English transl., Univ. of Chicago Press, Chicago, Ill., 1964. MR 13, 444; MR 28 #5038. MR 0161833 (28:5037)
  • [2] C. F. Osgood, On the Diophantine approximation of values of functions satisfying certain linear q-difference equations, J. Number Theory 3 (1971), 159-177. MR 0277481 (43:3214)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0308059-X
Keywords: Diophantine approximation, continued fractions, rational partial quotients
Article copyright: © Copyright 1972 American Mathematical Society

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