- Title
- Erdos properties of subsets of the Mahler set S
- Creator
- Chalebgwa, Taboka; Morris, Sidney
- Date
- 2023
- Type
- Text; Journal article
- Identifier
- http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/199444
- Identifier
- vital:19206
- Identifier
-
https://doi.org/10.1017/S0004972723000047
- Identifier
- ISSN:0004-9727 (ISSN)
- Abstract
- Erd.os proved that every real number is the sum of two Liouville numbers. A set W of complex numbers is said to have the Erd.os property if every real number is the sum of two members of W. Mahler divided the set of all transcendental numbers into three disjoint classes S, T and U such that, in particular, any two complex numbers which are algebraically dependent lie in the same class. The set of Liouville numbers is a proper subset of the set U and has Lebesgue measure zero. It is proved here, using a theorem of Weil on locally compact groups, that if m ∈ [0,∞), then there exist 2c dense subsets W of S each of Lebesgue measure m such that W has the Erd.os property and no two of these W are homeomorphic. It is also proved that there are 2c dense subsets W of S each of full Lebesgue measure, which have the Erd.os property. Finally, it is proved that there are 2c dense subsets W of S such that every complex number is the sum of two members of W and such that no two of these W are homeomorphic. © 2023 The Author(s).
- Publisher
- Cambridge University Press
- Relation
- Bulletin of the Australian Mathematical Society Vol. 108, no. 3 (2023), p. 504-510
- Rights
- All metadata describing materials held in, or linked to, the repository is freely available under a CC0 licence
- Rights
- https://creativecommons.org/licenses/by/4.0/
- Rights
- Copyright ©The Author(s), 2023
- Rights
- Open Access
- Subject
- 49 Mathematical sciences; Erdos properties; Lebesgue measure; Liouville number; Mahler classification; S-number; Transcendental number
- Full Text
- Reviewed
- Funder
- The first author’s research is supported by the Fields Institute for Research in Mathematical Sciences, via the Fields-Ontario postdoctoral Fellowship.
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