Furstenberg sets for a fractal set of directions
Authors:
Ursula Molter and Ezequiel Rela
Journal:
Proc. Amer. Math. Soc. 140 (2012), 27532765
MSC (2010):
Primary 28A78, 28A80
Published electronically:
December 1, 2011
MathSciNet review:
2910763
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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 , we will say that a set is an set if there is a subset of the unit circle of Hausdorff dimension at least and, for each direction in , there is a line segment in the direction of such that the Hausdorff dimension of the set is equal to or greater than . 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 for any . In particular we are able to extend previously known results to the ``endpoint'' case.
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Additional Information
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 – IMASUBA/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 – IMASUBA/CONICET, Argentina
Email:
erela@dm.uba.ar
DOI:
http://dx.doi.org/10.1090/S000299392011111110
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 PICT200600177, 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.
