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Infinitely repeated partitions of Liouville numbers

Part of: Elementary number theory Algebraic structures Sequences and sets

Published online by Cambridge University Press:  06 August 2025

Chungwu Ho Open the ORCID record for Chungwu Ho [Opens in a new window]  and
Seth Zimmerman Open the ORCID record for Seth Zimmerman [Opens in a new window]
Chungwu Ho*
Affiliation:
Department of Mathematics and Statistics, https://ror.org/04cqs5j56 Southern Illinois University at Edwardsville , Edwardsville, IL 62026, United States
Seth Zimmerman
Affiliation:
Department of Mathematics, Evergreen Valley College, San Jose, CA 95135, United States e-mail: zimls@earthlink.net
*
e-mail: cho@siue.edu
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Abstract

We show that the set of Liouville numbers has a rich set-theoretic structure: it can be partitioned in an explicit way into an uncountable collection of subsets, each of which is dense in the real line. Furthermore, each of these partitioning subsets can be similarly partitioned, and the process can be repeated indefinitely.

Keywords

Liouville numberspartitionsdense sets

MSC classification

Primary: 11B05: Density, gaps, topology
Secondary: 08A05: Structure theory 11A67: Other representations

Information

Type
Article
Information
Canadian Mathematical Bulletin , First View , pp. 1 - 8
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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