Required redundancy in the representation of reals

Starbird, Michael; Starbird, Thomas

doi:10.1090/S0002-9939-1992-1086343-5

Required redundancy in the representation of reals
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by Michael Starbird and Thomas Starbird PDF

Proc. Amer. Math. Soc. 114 (1992), 769-774 Request permission

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 \[ x = \sum \limits _{i = 0}^\infty {{a_i},} \] 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.

References

Marko Petkovšek, Ambiguous numbers are dense, Amer. Math. Monthly 97 (1990), no. 5, 408–411. MR 1048915, DOI 10.2307/2324393