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On the frequency distribution of partial quotients of U-numbers

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  26 February 2010

Edward B. Burger  and
Thomas Struppeck
Edward B. Burger
Affiliation:
Department of Mathematics, Williams College, Williamstown, Mass. 01267, U.S.A.
Thomas Struppeck
Affiliation:
Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712, U.S.A.
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Extract

In Mahler's classification of complex numbers [10] (see [4]), a transcendental number ξ is called a U-number if there exists a fixed integer N ≥ 1 so that for all ω > 0, there exists a polynomial so that

where the height h(f) = max {|α0|, |α1|, …, |αN|}. The number ξ is called a Um-number if the above holds for N = m but for no smaller value of N (examples and further details may be found in [9,1 and 2]). Thus the set of U1-numbers is precisely the set of Liouville numbers. In this paper we investigate the statistical behavior of the partial quotients of real U-numbers, in particular, U2-numbers. In addition, we demonstrate the existence of a U2-number with the property that if it is translated by any nonnegative integer and then squared, the result is a Liouville number. Related results involving badly approximate U2-numbers are also discussed.

MSC classification

Secondary: 11J82: Measures of irrationality and of transcendence
Type
Research Article
Information
Mathematika , Volume 40 , Issue 2 , December 1993 , pp. 215 - 225
Copyright
Copyright © University College London 1993

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