Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 903-923

HAUSDORFF DIMENSION OF FURSTENBERG-TYPE SETS ASSOCIATED TO FAMILIES OF AFFINE SUBSPACES

Kornélia Héra

Eötvös Loránd University, Institute of Mathematics
Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary; herakornelia 'at' gmail.com

Abstract. We show that if B ⊂ Rⁿ and E ⊂ A(n,k) is a nonempty collection of k-dimensional affine subspaces of Rⁿ such that every P ∈ E intersects B in a set of Hausdorff dimension at least α with k – 1 < α ≤ k, then dim B ≥ α + dim E/(k + 1), where dim denotes the Hausdorff dimension. This estimate generalizes the well known Furstenberg-type estimate that every α-Furstenberg set in the plane has Hausdorff dimension at least α + 1/2. More generally, we prove that if B and E are as above with 0 < α ≤ k, then dim B ≥ α + (dim E – (k– ⌈α⌉)(n – k))/(⌈α⌉ + 1). We also show that this bound is sharp for some parameters. As a consequence, we prove that for any 1 ≤ k < n, the union of any nonempty s-Hausdorff dimensional family of k-dimensional affine subspaces of Rⁿ has Hausdorff dimension at least k + s/(k + 1).

2010 Mathematics Subject Classification: Primary 28A78, 05B30.

Key words: Hausdorff dimension, affine subspaces, Furstenberg sets.

Reference to this article: K. Héra: Hausdorff dimension of Furstenberg-type sets associated to families of affine subspaces. Ann. Acad. Sci. Fenn. Math. 44 (2019), 903-923.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4469