Hausdorff dimension of unions of affine subspaces and of Furstenberg-type sets
Kornélia Héra
Eötvös Loránd University, Budapest, HungaryTamás Keleti
Eötvös Loránd University, Budapest, HungaryAndrás Máthé
University of Warwick, Coventry, UK
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Abstract
We prove that for any and , the union of any nonempty -Hausdorff dimensional family of -dimensional affine subspaces of has Hausdorff dimension . More generally, we show that for any , if and is a nonempty collection of -dimensional affine subspaces of such that every intersects in a set of Hausdorff dimension at least , then dim , where dim denotes the Hausdorff dimension. As a consequence, we generalize the well-known Furstenberg-type estimate that every -Furstenberg set has Hausdorff dimension at least ; we strengthen a theorem of Falconer and Mattila [5]; and we show that for any , if a set contains the -skeleton of a rotated unit cube around every point of , or if contains a -dimensional affine subspace at a fixed positive distance from every point of , then the Hausdorff dimension of is at least .
Cite this article
Kornélia Héra, Tamás Keleti, András Máthé, Hausdorff dimension of unions of affine subspaces and of Furstenberg-type sets. J. Fractal Geom. 6 (2019), no. 3, pp. 263–284
DOI 10.4171/JFG/77