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Formulas for some diophantine approximation constants

Published online by Cambridge University Press:  09 April 2009

T. W. Cusick  and
S. Krass
T. W. Cusick
Affiliation:
Department of MathematicsState University of New YorkBuffalo, New York 14214, U.S.A.
S. Krass
Affiliation:
School of MathematicsUniversity of New South WalesKensington, New South Wales 2033, Australia
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Abstract

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Let M be a full Z-module in F a real number field of degree at least 3 with N(α) denoting the norm of α ∈ F. Given any nonzero number φ in M we make the plausible conjecture that one can find a number ß in M such that N(ß) = N(φ) and the algebraic conjugates of ß (not including ß) have ratios arbitraily near any given numbers consistent with the complex algebraic conjugates of elements of F. We use the conjecture to give explicit formulas for some diophantine approximation constants. Without the conjecture our methods lead to corresponding lower bounds for these constants.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 44 , Issue 3 , June 1988 , pp. 311 - 323
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Adams, W. W., ‘Simultaneous asymptotic diophantine approximations to a basis of a real cubic number field’, J. Number Theory 1 (1969), 179–194.CrossRefGoogle Scholar
[2]Ax, J., ‘On Schanuel's Conjecture’, Ann. of. Math. 93 (1971), 252–268.CrossRefGoogle Scholar
[3]Cusick, T. W., ‘Formulas for some diophantine approximation constants, II’, Acta Arithmetica 26 (1974–1975), 117–128.CrossRefGoogle Scholar
[4]Krass, S., The n-dimensional diophantine approximation constants (Ph. D. Thesis, University of New South Wales, 1984).Google Scholar
[5]Krass, S., ‘On a conjecture of Littlewood in diophantine approximations’, Bull. Austral. Math. Soc. 32 (1985), 379–387.CrossRefGoogle Scholar
[6]Peck, L. G., ‘Simultaneous rational approximations to algebraic numbers’, Bull. Amer. Math. Soc. 67 (1961), 197–201.CrossRefGoogle Scholar