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Proceedings of the American Mathematical Society

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



Required redundancy in the representation of reals

Authors: Michael Starbird and Thomas Starbird
Journal: Proc. Amer. Math. Soc. 114 (1992), 769-774
MSC: Primary 40A05
MathSciNet review: 1086343
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Abstract: Redundancy in decimal-like representations of reals cannot be avoided. It is proved here that if $ {\{ {A_i}\} _{i = 0,1,2, \ldots }}$ is a countable collection of countable (or finite) sets of reals such that for each real $ x$ there are $ {a_i} \in {A_i}$ with

$\displaystyle x = \sum\limits_{i = 0}^\infty {{a_i},} $

then there is a dense subset of reals with redundant representations; that is, there is a dense set $ C$ of $ \mathbb{R}$ such that for each $ x$ in $ C,\quad x = \sum\nolimits_{i = 0}^\infty {{a_i}} $ and $ x = \sum\nolimits_{i = 0}^\infty {{b_i}} $ with $ {a_i}\,,\quad {b_i}$ in $ {A_i}$, but $ {a_i} \ne {b_i}$ for some $ i$. Petkovsek [1] proved a similar result under the added assumption that every sum of the form $ \sum\nolimits_{i = 0}^\infty {{a_i}} $ with $ {a_i} \in {A_i}$ converges.

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Keywords: Ambiguous numbers, redundant representations
Article copyright: © Copyright 1992 American Mathematical Society

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