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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 α, β ∈ (0, 1], we will say that a set E⊂ℝ 2 is an F αβ-set if there is a subset L of the unit circle of Hausdorff dimension at least β and, for each direction e in L, there is a line segment ℓ e in the direction of e such that the Hausdorff dimension of the set E∩ℓ e is equal to or greater than α. 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) ≥ max {α + β/2; 2α + β-1} for any E ∈ F αβ. In particular we are able to extend previously known results to the "endpoint" α = 0 case. © 2011 American Mathematical Society.

Registro:

Documento: Artículo
Título:Furstenberg sets for a fractal set of directions
Autor:Molter, U.; Rela, E.
Filiación:Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I, 1428 Capital Federal, Argentina
IMAS-UBA/CONICET, Argentina
Palabras clave:Dimension function; Furstenberg sets; Hausdorff dimension; Kakeya sets
Año:2012
Volumen:140
Número:8
Página de inicio:2753
Página de fin:2765
DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11111-0
Título revista:Proceedings of the American Mathematical Society
Título revista abreviado:Proc. Am. Math. Soc.
ISSN:00029939
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_00029939_v140_n8_p2753_Molter

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Citas:

---------- APA ----------
Molter, U. & Rela, E. (2012) . Furstenberg sets for a fractal set of directions. Proceedings of the American Mathematical Society, 140(8), 2753-2765.
http://dx.doi.org/10.1090/S0002-9939-2011-11111-0
---------- CHICAGO ----------
Molter, U., Rela, E. "Furstenberg sets for a fractal set of directions" . Proceedings of the American Mathematical Society 140, no. 8 (2012) : 2753-2765.
http://dx.doi.org/10.1090/S0002-9939-2011-11111-0
---------- MLA ----------
Molter, U., Rela, E. "Furstenberg sets for a fractal set of directions" . Proceedings of the American Mathematical Society, vol. 140, no. 8, 2012, pp. 2753-2765.
http://dx.doi.org/10.1090/S0002-9939-2011-11111-0
---------- VANCOUVER ----------
Molter, U., Rela, E. Furstenberg sets for a fractal set of directions. Proc. Am. Math. Soc. 2012;140(8):2753-2765.
http://dx.doi.org/10.1090/S0002-9939-2011-11111-0