Artículo
Furstenberg sets for a fractal set of directions
Fecha de publicación:
12/2012
Editorial:
American Mathematical Society
Revista:
Proceedings of the American Mathematical Society
ISSN:
0002-9939
Idioma:
Inglés
Tipo de recurso:
Artículo publicado
Clasificación temática:
Resumen
In this note 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 $alpha,etain(0,1]$, we will say that a set $Esubset R^2$ is an $F_{alphaeta}$-set if there is a subset $L$ of the unit circle of Hausdorff dimension at least $eta$ 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 $Ecapell_e$ is equal 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 $dim(E)gemaxleft{alpha+rac{eta}{2} ; 2alpha+eta -1 ight}$ for any $Ein F_{alphaeta}$. In particular we are able to extend previously known results to the ``endpoint´´ $alpha=0$ case.
Palabras clave:
HAUSDORFF DIMENSION
,
FURSTENBERG SET
,
KAKEYA SET
,
DIMENSION FUNCTION
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Articulos(IMAS)
Articulos de INSTITUTO DE INVESTIGACIONES MATEMATICAS "LUIS A. SANTALO"
Articulos de INSTITUTO DE INVESTIGACIONES MATEMATICAS "LUIS A. SANTALO"
Citación
Molter, Ursula Maria; Rela, Ezequiel; Furstenberg sets for a fractal set of directions; American Mathematical Society; Proceedings of the American Mathematical Society; 140; 8; 12-2012; 2753-2765
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