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Rational approximations to algebraic numbers

Published online by Cambridge University Press:  26 February 2010

H. Davenport  and
K. F. Roth
H. Davenport
Affiliation:
University College, London.
K. F. Roth
Affiliation:
University College, London.
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Extract

It was proved in a recent paper that if α is any algebraic number, not rational, then for any ζ > 0 the inequality

has only a finite number of solutions in relatively prime integers h, q. Our main purpose in the present note is to deduce, from the results of that paper, an explicit estimate for the number of solutions.

Type
Research Article
Information
Mathematika , Volume 2 , Issue 2 , December 1955 , pp. 160 - 167
Copyright
Copyright © University College London 1955

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References

page 160 note * Roth, K. F., “Rational approximations to algebraic numbers”, Mathematika 2 (1955), 1–20CrossRefGoogle Scholar. This paper will be referred to as R.

page 160 note † We exclude the case n = 2 because it can be treated more effectively by much simpler and well-known methods.

page 162 note * It was supposed in R that α is real, merely because the result of that paper is trivially true otherwise. All the details of R remain valid if α is complex, and it is convenient not to restrict α to be real in the present work, in order to avoid a minor complication in the proof of Theorem 2.

page 164 note * Perron, , Algebra II (Berlin 1951), Satz 18Google Scholar.