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

 

Furstenberg sets for a fractal set of directions


Authors: Ursula Molter and Ezequiel Rela
Journal: Proc. Amer. Math. Soc. 140 (2012), 2753-2765
MSC (2010): Primary 28A78, 28A80
Published electronically: December 1, 2011
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Abstract: In this paper we study the behavior of the size of Furstenberg sets with respect to the size of the set of directions defining it. For any pair $ \alpha ,\beta \in (0,1]$, we will say that a set $ E\subset \mathbb{R}^2$ is an $ F_{\alpha \beta }$-set if there is a subset $ L$ of the unit circle of Hausdorff dimension at least $ \beta $ and, for each direction $ e$ in $ L$, there is a line segment $ \ell _e$ in the direction of $ e$ such that the Hausdorff dimension of the set $ E\cap \ell _e$ is equal to or greater than $ \alpha $. The problem is considered in the wider scenario of generalized Hausdorff measures, giving estimates on the appropriate dimension functions for each class of Furstenberg sets. As a corollary of our main results, we obtain that $ \dim (E)\ge \max \left \{\alpha +\frac {\beta }{2} ; 2\alpha +\beta -1\right \}$ for any $ E\in F_{\alpha \beta }$. In particular we are able to extend previously known results to the ``endpoint'' $ \alpha =0$ case.


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Ursula Molter
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Capital Federal, Argentina – and – IMAS-UBA/CONICET, Argentina
Email: umolter@dm.uba.ar

Ezequiel Rela
Affiliation: Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Capital Federal, Argentina – and – IMAS-UBA/CONICET, Argentina
Email: erela@dm.uba.ar

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11111-0
PII: S 0002-9939(2011)11111-0
Keywords: Furstenberg sets, Hausdorff dimension, dimension function, Kakeya sets
Received by editor(s): September 2, 2010
Received by editor(s) in revised form: March 6, 2011
Published electronically: December 1, 2011
Additional Notes: This research is partially supported by grants ANPCyT PICT2006-00177, CONICET PIP 11220080100398 and UBACyT X149
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.