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On numbers with many rational approximations

Published online by Cambridge University Press:  24 October 2008

R. C. Baker
R. C. Baker
Affiliation:
Royal Holloway College, Egham, Surrey
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Extract

Every real number α admits many rational approximations in the sense that for every natural number q,

for some integer p. Perhaps surprisingly, numbers α which have k distinct rational approximations with given denominators q1,…,qk,

for a fixed ε > 0, are quite sparse; their measure in U = [0,1) is at most C(ε)/k.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 86 , Issue 1 , July 1979 , pp. 25 - 27
Copyright
Copyright © Cambridge Philosophical Society 1979

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References

REFERENCE

(1)Vinogradov, I. M.The method of trigonometric sums in the theory of numbers (London, New York, Interscience, 1954).Google Scholar