Hausdorff dimension of unions of affine subspaces and of Furstenberg-type sets

  • Kornélia Héra

    Eötvös Loránd University, Budapest, Hungary
  • Tamás Keleti

    Eötvös Loránd University, Budapest, Hungary
  • András Máthé

    University of Warwick, Coventry, UK
Hausdorff dimension of unions of affine subspaces and of Furstenberg-type sets cover

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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