Required redundancy in the representation of reals
HTML articles powered by AMS MathViewer
- 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
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 114 (1992), 769-774
- MSC: Primary 40A05
- DOI: https://doi.org/10.1090/S0002-9939-1992-1086343-5
- MathSciNet review: 1086343