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Herbrand-Analysen zweier Beweise des Satzes von Roth: Polynomiale Anzahlschranken

Published online by Cambridge University Press:  12 March 2014

H. Luckhardt
H. Luckhardt*
Affiliation:
Fachbereich Mathematik, Johann Wolfgang Goethe-Universität, 6000 Frankfurt am Main, West Germany
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Abstract

A previously unexplored method, combining logical and mathematical elements, is shown to yield substantial numerical improvements in the area of Diophantine approximations. Kreisel illustrated the method abstractly by noting that effective bounds on the number of elements are ensured if Herbrand terms from ineffective proofs of Σ2-finiteness theorems satisfy certain simple growth conditions. Here several efficient growth conditions for the same purpose are presented that are actually satisfied in practice, in particular, by the proofs of Roth's theorem due to Roth himself and to Esnault and Viehweg. The analysis of the former yields an exponential bound of order exp(70ε−2d2) in place of exp(285ε−2d2) given by Davenport and Roth in 1955, where α is (real) algebraic of degree d ≥ 2 and ∣α − pq−1∣ < q−2−ε. (Thus the new bound is less than the fourth root of the old one.) The new bounds extracted from the other proof are polynomial of low degree (in ε−1 and log d). Corollaries: Apart from a new bound for the number of solutions of the corresponding Diophantine equations and inequalities (among them Thue's inequality), log log qν, < Cα, εν5/6+ε, where qν are the denominators of the convergents to the continued fraction of α.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 1 , March 1989 , pp. 234 - 263
Copyright
Copyright © Association for Symbolic Logic 1989

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References

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