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Rational approximations to algebraic numbers

  • H. Davenport (a1) and K. F. Roth (a1)

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.

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page 160 note * Roth, K. F., “Rational approximations to algebraic numbers”, Mathematika 2 (1955), 120. 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 18.

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  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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