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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On Liouville decompositions in local fields

Author: Edward B. Burger
Journal: Proc. Amer. Math. Soc. 124 (1996), 3305-3310
MSC (1991): Primary 11J61, 11J81
MathSciNet review: 1350935
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Abstract: In 1962 Erd\H{o}s proved that every real number may be decomposed into a sum of Liouville numbers. Here we consider more general functions which decompose elements from an arbitrary local field into Liouville numbers. Several examples and applications are given. As an illustration, we prove that for any real numbers $\alpha _{1},\thinspace \alpha _{2},\ldots ,\thinspace \alpha _{N}$, not equal to 0 or 1, there exist uncountably many Liouville numbers $\sigma $ such that $\alpha _{1}^{\sigma },\thinspace \alpha _{2}^{\sigma },\thinspace \ldots ,\thinspace \alpha _{N}^{\sigma }$ are all Liouville numbers.

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Edward B. Burger
Affiliation: Department of Mathematics, Williams College, Williamstown, Massachusetts 01267

Received by editor(s): April 6, 1994
Received by editor(s) in revised form: May 3, 1995
Communicated by: William W. Adams
Article copyright: © Copyright 1996 American Mathematical Society