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

 

Dirichlet sets, Erdős-Kunen-Mauldin theorem, and analytic subgroups of the reals


Author: Peter Eliaš
Journal: Proc. Amer. Math. Soc. 139 (2011), 2093-2104
MSC (2010): Primary 28A05, 54H05, 54H11
Published electronically: November 10, 2010
MathSciNet review: 2775387
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Abstract: We prove strengthenings of two well-known theorems related to the Lebesgue measure and additive structure of the real line. The first one is a theorem of Erdős, Kunen, and Mauldin stating that for every perfect set there exists a perfect set of measure zero such that their algebraic sum is the whole real line. The other is Laczkovich's theorem saying that every proper analytic subgroup of the real line is included in an $ F_\sigma$ set of measure zero. Using the strengthened theorems we generalize the fact that permitted sets for families of trigonometric thin sets are perfectly meager.


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Additional Information

Peter Eliaš
Affiliation: Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, Košice, Slovakia
Email: elias@upjs.sk

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10639-1
PII: S 0002-9939(2010)10639-1
Keywords: Dirichlet sets, permitted sets, analytic subgroups of the reals, Kronecker’s theorem, Erdős-Kunen-Mauldin theorem
Received by editor(s): September 10, 2008
Received by editor(s) in revised form: June 4, 2010
Published electronically: November 10, 2010
Additional Notes: This work was supported by grant No. 1/0032/09 of Slovak Grant Agency VEGA
Communicated by: Julia Knight
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.