Dirichlet sets, Erdős-Kunen-Mauldin theorem, and analytic subgroups of the reals
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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.References
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Additional Information
- Peter Eliaš
- Affiliation: Mathematical Institute, Slovak Academy of Sciences, Grešákova 6, Košice, Slovakia
- Email: elias@upjs.sk
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 2093-2104
- MSC (2010): Primary 28A05, 54H05, 54H11
- DOI: https://doi.org/10.1090/S0002-9939-2010-10639-1
- MathSciNet review: 2775387