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On the Hausdorff dimension of pinned distance sets
(Hebrew Univ Magnes Press, 2019-04-17)
We prove that if A is a Borel set in the plane of equal Hausdorff and packing dimension s > 1, then the set of pinned distances {|x − y| : y ∈ A} has full Hausdorff dimension for all x outside of a set of Hausdorff dimension ...
Irrationality exponent, Hausdorff dimension and effectivization
(Springer Wien, 2018-02)
We generalize the classical theorem by Jarnik and Besicovitch on the irrationality exponents of real numbers and Hausdorff dimension and show that the two notions are independent. For any real number a greater than or equal ...
Finite (Hausdorff) dimension of plants and roots as indicator of ontogenyDimensión finita (de Hausdorff) de plantas y raíces como indicador de ontogenia
(Universidad Nacional de Cuyo, 2019-12)
La arquitectura de las plantas responde a procesos endógenos y a la influencia de factores ambientales. El estudio alométrico de la arquitectura ha sido un desafío para los biólogos. En este trabajo definimos una nueva ...
Improved bounds for the dimensions of planar distance sets
(European Mathematical Society, 2020-12)
We obtain new lower bounds on the Hausdorff dimension of distance sets and pinned distance sets of planar Borel sets of dimension slightly larger than 1, improving recent estimates of Keleti and Shmerkin, and of Liu in ...
Squares and their centers
(Springer, 2018-02)
We study the relationship between the size of two sets B, S ⊂ R2, when B contains either the whole boundary or the four vertices of a square with axes-parallel sides and center in every point of S. Size refers to cardinality, ...
New bounds on the dimensions of planar distance sets
(Birkhauser Verlag Ag, 2019-07)
We prove new bounds on the dimensions of distance sets and pinned distance sets of planar sets. Among other results, we show that if A ⊂ R2 is a Borel set of Hausdorff dimension s > 1, then its distance set has Hausdorff ...
Improved bounds on the dimensions of sets that avoid approximate arithmetic progressions
(Birkhauser Boston Inc, 2021-02)
We provide quantitative estimates for the supremum of the Hausdorff dimension of sets in the real line which avoid ε-approximations of arithmetic progressions. Some of these estimates are in terms of Szemerédi bounds. In ...
Classifying cantor sets by their fractal dimensions
(American Mathematical Society, 2010-11)
In this article we study Cantor sets defined by monotone sequences, in the sense of Besicovich and Taylor. We classify these Cantor sets in terms of their h-Hausdorff and h-packing measures, for the family of dimension ...
Improving dimension estimates for Furstenberg-type sets
(Academic Press Inc Elsevier Science, 2010-01)
In this paper we study the problem of estimating the generalized Hausdorff dimension of Furstenberg sets in the plane. For α ∈ (0, 1], a set F in the plane is said to be an α-Furstenberg set if for each direction e there ...
Dimension functions of Cantor sets
(American Mathematical Society, 2007-10)
We estimate the packing measure of Cantor sets associated to nonincreasing sequences through their decay. This result, dual to one obtained by Besicovitch and Taylor, allows us to characterize the dimension functions ...