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A surprising covering of the real line

Author(s): Gábor Kun
Journal: Proc. Amer. Math. Soc. 134 (2006), 3555-3559.
MSC (2000): Primary 28A05, 03E15
Posted: June 8, 2006
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Abstract: We construct an increasing sequence of Borel subsets of $\mathbb{R}$ , such that their union is $\mathbb{R}$ , but $\mathbb{R}$ cannot be covered with countably many translations of one set. The proof uses a random method.

References:

1.: M. Abért and T. Keleti, Shuffle the plane, Proc. Amer. Math. Soc. 130, (2002), 549-553. MR 1862136 (2003g:03070)
2.: Udayan B. Darji and Tamás Keleti, Covering $\mathbb{R}$ with translates of a compact set, Proc. Amer. Math. Soc. 131 (2003), 2593-2596. MR 1974660 (2004d:03100)
3.: P. Erdos, Some remarks on set theory, Annals of Math. 44, (1943), 643-646.MR 0009614 (5:173c)
4.: Z. Ruzsa, Euklideszi terek fedése kis halmazokkal (in Hungarian), Thesis for the Master`s Degree (2001).
5.: P. Komjáth, Five degrees of separation, Proc. Amer. Math. Soc. 130, (2002), 2413-2417. MR 1897467 (2003c:03082)

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Additional Information:

Gábor Kun
Affiliation: Department of Algebra and Number Theory, Eötvös Loránd University, 1117 Pázmány Péter sétány 1/c, Budapest, Hungary
Email: kungabor@cs.elte.hu

DOI: 10.1090/S0002-9939-06-08371-7
PII: S 0002-9939(06)08371-7
Keywords: Borel set, translates, countable, covering
Received by editor(s): September 23, 2003
Received by editor(s) in revised form: November 10, 2004 and June 17, 2005
Posted: June 8, 2006
Additional Notes: The research of the author was supported by OTKA grant no. T032042 and T049786. The author is indebted to Z. Ruzsa for his helpful remarks.
Communicated by: Carl G. Jockusch, Jr.
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.


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