J Austral Math Soc Ser A 44 pp311--323, 1988.

Formulas for Some Diophantine Approximation Constants

T. W. Cusick and S. Krass

Abstract

Let M be a full Z-module if F a real number field of degree at least 3 with N(a) denoting the norm of a Î F. Given any nonzero number f in M we make the plausible conjecture that one can find a number b in M such that N(b) = N(f) and the algebraic conjugates of b (not including b) have ratios arbitrarily 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 ourr methods lead to corresponding lower bounds for these constants.

1980 AMS Subject Classification: 10F10, 10F25

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Authors

T. W. Cusick

Department of Mathematics, State University of New York, Buffalo, New York 14214, U.S.A.

S. Krass

School of Mathematics, University of New South Wales, Kensington, New South Wales 2033, Australia.