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Required Redundancy in the Representation of Reals

Michael Starbird and Thomas Starbird
Proceedings of the American Mathematical Society
Vol. 114, No. 3 (Mar., 1992), pp. 769-774
DOI: 10.2307/2159403
Stable URL: http://www.jstor.org/stable/2159403
Page Count: 6
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Required Redundancy in the Representation of Reals
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Abstract

Redundancy in decimal-like representations of reals cannot be avoided. It is proved here that if {Ai}i = 0, 1, 2,... is a countable collection of countable (or finite) sets of reals such that for each real x there are ai ∈ Ai with x = ∑∞ i = 0ai, then there is a dense subset of reals with redundant representations; that is, there is a dense set C of R such that for each x in C, x = ∑∞ i = 0ai and x = ∑∞ i = 0bi with ai, bi in Ai, but ai ≠ bi for some i. Petkovsek [1] proved a similar result under the added assumption that every sum of the form ∑∞ i = 0ai with ai ∈ Ai converges.

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