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Roth’s Theorem

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Part of the Lecture Notes in Mathematics book series (LNM, volume 785)

Abstract

THEOREM 1A (Liouville (1844)). Suppose α is a real algebraic number of degree d. Then there is a constant c(α) > 0 such that
$$ \left| {\alpha - \frac{p} {q}} \right| > \frac{{c(\alpha )}} {{q^d }} $$
for every rational \( \tfrac{p} {q} \) distinct from α.

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References

  1. A. Thue (1908). Om en generel i store hele taluløsbar ligning. Kra. Vidensk. Selsk. Skrifter. I. Mat. Nat. Kl. No. 7. Kra.Google Scholar
  2. ____ (1909). Über Annäherungswerte algebraischer Zahlen. J. reine ang. Math. 135, 284–305.Google Scholar
  3. K.F. Roth (1955a). Rational approximations to algebraic numbers. Mathematika 2, 1–20.MathSciNetCrossRefGoogle Scholar
  4. W.M. Schmidt (1971d). Approximation to algebraic numbers. L’Enseignement Math. 17, 187–253.zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1980

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