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Rational approximations to algebraic numbers

  • K. F. Roth (a1)
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It was remarked by Liouville in 1844 that there is an obvious limit to the accuracy with which algebraic numbers can be approximated by rational numbers; if α is an algebraic number of degree n (at least 2) then

for all rational numbers h/q, where A is a positive number depending only on α.

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page 1 note † The result is an immediate deduction from the definition of an algebraic number; see, for example, Davenport, , The Higher Arithmetic (London 1952), 165167.

page 1 note ‡ Acta Mathematica, 79 (1947), 225240. The algebraic part of Dyson's work was simplified by Mahler, , Proc. K. Akad. Wet. Amsterdam, 52 (1949), 11751184. Another proof of Dyson's result was given by Schneider, in Archiv der Math., 1 (1948–9), 288295. Dyson's result (with a generalization) was apparently obtained independently by Gelfond; see his Transcendental and algebraic numbers (Moscow 1952, in Russian), Chapter 1.

page 2 note † See Skolem, , Diophantische Gleichungen (Ergebnisse der Math. V4, Berlin, 1938), Chapter 6, §2.

page 2 note ‡ Math. Zeitschrift, 9 (1921), 173213.

page 3 note † J. für die reine und angew. Math., 175 (1936), 182192, Lemma 1, formula (7). This paper contains a proof that κ ≤ 2 provided that the solutions of (1) satisfy a certain very restrictive condition.

page 3 note ‡ Since writing this paper I find that generalized Wronskians were used by Siegel [Math. Annalen, 84 (1921), 8099] in a similar connection. See also Kellogg, , Comptes rendus des séances de la Soc. Math. de France, 41 (1912), 1921, where the main result (Lemma 1 below) is stated without proof.

page 3 note § It should perhaps be remarked (though it is immaterial to our argument) that the generalized Wronskians and their derivatives may satisfy identities, by virtue of which the vanishing of some of the generalized Wronskians implies the vanishing of the others,

page 6 note † See, for example, Perron, , Algebra I (Berlin, 1927, 1931, 1951), Satz 88. The deduction does not depend on the separation of the variables between G and W.

page 9 note † The exponent θr 1 is of course a non-negative integer, and can be supposed to be a positive integer.

page 14 note † The case of even r would in fact suffice for the application later, since we could choose r 1, …, r m in §8 so as to be even.

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