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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Borel subrings of the reals


Authors: G. A. Edgar and Chris Miller
Journal: Proc. Amer. Math. Soc. 131 (2003), 1121-1129
MSC (2000): Primary 28A78; Secondary 03E15, 11K55, 12D99, 28A05
Published electronically: June 12, 2002
MathSciNet review: 1948103
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Abstract: A Borel (or even analytic) subring of $\mathbb R$ either has Hausdorff dimension $0$ or is all of $\mathbb R$. Extensions of the method of proof yield (among other things) that any analytic subring of $\mathbb C$ having positive Hausdorff dimension is equal to either $\mathbb R$ or $\mathbb C$.


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Additional Information

G. A. Edgar
Affiliation: Department of Mathematics, The Ohio State University, 231 West Eighteenth Avenue, Columbus, Ohio 43210
Email: edgar@math.ohio-state.edu

Chris Miller
Affiliation: Department of Mathematics, The Ohio State University, 231 West Eighteenth Avenue, Columbus, Ohio 43210
Email: miller@math.ohio-state.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06653-4
PII: S 0002-9939(02)06653-4
Keywords: Borel subring, Borel subfield, Hausdorff dimension, Erd\H{o}s, Volkmann, Suslin sets, analytic sets
Received by editor(s): October 29, 2001
Published electronically: June 12, 2002
Additional Notes: Research of the second author was supported by NSF grant no. DMS-9988855
Communicated by: David Preiss
Article copyright: © Copyright 2002 American Mathematical Society