2018/09/12 by Kornélia Héra, Héra, Kornélia · 2 citations
Mathematics · #Advanced Topology and Set Theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.1809.04666
openalex publication_date 2018/09/12 · openalex created_date 2022/08/03 · openalex updated_date 2026/07/28
We show that if B \⊂ \ℝn and E \⊂ A(n,k) is a nonempty\ncollection of k-dimensional affine subspaces of \ℝn such that\nevery P \∈ E intersects B in a set of Hausdorff dimension at least\n\α with k-1 < \α \≤ k, then \dim B \≥ \α +\dim E/(k+1),\nwhere \dim denotes the Hausdorff dimension. This estimate generalizes the\nwell known Furstenberg-type estimate that every \α-Furstenberg set in the\nplane has Hausdorff dimension at least \α + 1/2.\n More generally, we prove that if B and E are as above with 0 < \α\n\≤ k, then \dim B \≥ \α +(\dim E-(k- lceil \α\n rceil)(n-k))/( lceil \α rceil+1). We also show that this bound is sharp\nfor some parameters.\n As a consequence, we prove that for any 1 \≤ k<n, the union of any\nnonempty s-Hausdorff dimensional family of k-dimensional affine subspaces\nof \ℝn has Hausdorff dimension at least k+\(s)/(k+1).\n