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Simultaneous Diophantine Approximation

Published online by Cambridge University Press:  26 February 2010

H. Davenport
H. Davenport
Affiliation:
University College, London, W.C.I.
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Extract

It is well known that every irrational number θ possesses an infinity of rational approximations p/q satisfying

It is also well known that there is a wide class of irrational numbers which admit of no approximations which are essentially better, namely those θ whose continued fractions have bounded partial quotients. For any such θ there is a positive number c such that all rational approximations satisfy

Type
Research Article
Information
Mathematika , Volume 1 , Issue 1 , June 1954 , pp. 51 - 72
Copyright
Copyright © University College London 1954

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References

page note 51 * Math. Annalen, 83 (1921), 77à84.

page note 52 * This is a special ease of the “Uebertragungsprinzip”: see Koksma, , Diophantische Approximationen (Berlin, 1936), ch. 5, §1.Google Scholar

page note 53 * The field k is the cubic subfleld of the field of the 7th roots of unity, and α = 2 cos 2π/7, β = 2 cos 4π/7, γ = 2 cos 8π/7.

page note 53 † The field K is the cubic subfield of the field of the 7th roots of unity, and A = - 2 cos 8α/9, B = – 2 cos 4α/9, r = - 2 cos 2π/9.

page note 66 * An alternative method is to use the formulae of §6. We can express τn+1(1+p n+1), τn+1q n+1, τn+1r n+1 linearly in terms of 1+p n, q n, r n as in (42), (43), (44). The inverse formulae lead to the same expressions for qn and r n as those obtained above.