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On approximation with algebraic numbers of bounded degree

Published online by Cambridge University Press:  26 February 2010

R. C. Baker
R. C. Baker
Affiliation:
Royal Holloway College, Egham, Surrey.
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Extract

In this paper we are interested in two related measures of the degree of approximation of a complex number ζ by algebraic numbers. For a given integer n ≥ 1, write wn(ζ) for the supremum of the exponents w for which

for infinitely many polynomials

in Z[x] of height H(P) = max |av|. Clearly 0 ≤ w1 (ζ) ≤ w2 (ζ) ≤ …. On the other hand, write for the supremum of the exponents w for which

for infinitely many algebraic numbers α of degree at most n.

Type
Research Article
Information
Mathematika , Volume 23 , Issue 1 , June 1976 , pp. 18 - 31
Copyright
Copyright © University College London 1976

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References

1.Herstein, I. N.. Topics in Algebra (Xerox, Lexington 1964).Google Scholar
2.Koksma, J. F.. “Ober die Mahlersche Klasseinteilung der transzendenten Zahlen und die Approximation komplexer Zahlen durch algebraische Zahlen”, Monatsh. Math. Phys., 48 (1939), 176–189.CrossRefGoogle Scholar
3.Mahler, K.. “Zur Approximation der Exponential-funktion und des Logarithmus I, II”. J. reine angew. Math., 166 (1932), 118–150.CrossRefGoogle Scholar
4.Mahler, K.. “An application of Jensen's formula to polynomials”, Mathematika, 7 (1960), 98–100.CrossRefGoogle Scholar
5.Schmidt, W. M.. “T-numbers do exist”, Symposia Math. IV. INDAM, Rome, 1968 (Academic Press, London, 1970), 3–26.Google Scholar
6.Schmidt, W. M.. “Mahler's T-numbers”, Proc. Symposia Pure Math., vol. 20 (Amer. Math. Soc. 1971), 275–286.CrossRefGoogle Scholar
7.Schmidt, W. M.. “Simultaneous approximation to algebraic numbers by rationals”, Acta Math., 125 (1970), 189–201.CrossRefGoogle Scholar
8.Wirsing, E.. “Approximation mit algebraischen Zahlen beschrankten Grades”, J. reine angew. Math., 206 (1961), 67–77.CrossRefGoogle Scholar
9.Baker, A. and Schmidt, W. M.. “Diophantine approximation and Hausdorff dimension”, Proc. Lond. Math. Soc., 21 (1970), 1–11.CrossRefGoogle Scholar
10.Wirsing, E.. “On approximations of algebraic numbers by algebraic numbers of bounded degree”, Proc. Symposia Pure Math. vol. 20 (Amer. Math. Soc, 1971), 213–247.CrossRefGoogle Scholar