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

Abstract

We prove that for any 1 ≤ k < n and s ≤ 1, the union of any nonempty s-Hausdorff dimensional family of k-dimensional affine subspaces of R n has Hausdorff dimension k + s. More generally, we show that for any 0 < α ≤ k, if B ⊂ R n and E is a nonempty collection of k-dimensional affine subspaces of R n such that every P ∈ E intersects B in a set of Hausdorff dimension at least α, then dim B ≥ 2α − k + min(dim E, 1), where dim denotes the Hausdorff dimension. As a consequence, we generalize the well-known Furstenberg-type estimate that every α-Furstenberg set has Hausdorff dimension at least 2α; we strengthen a theorem of Falconer and Mattila [5]; and we show that for any 0 ≤ k < n, if a set A ⊂ R n contains the k-skeleton of a rotated unit cube around every point of R n , or if A contains a k-dimensional affine subspace at a fixed positive distance from every point of R n , then the Hausdorff dimension of A is at least k + 1.

Hausdorff dimension of Furstenberg-type sets associated to families of affine subspaces

“…In [5] the authors investigated Furstenberg-type sets associated to families of affine subspaces. For any integers 1 ≤ k < n, let A(n, k) denote the space of all k-dimensional affine subspaces of R n .…”

“…We say that B ⊂ R n is an (α, k, s)-Furstenberg set, if there is ∅ = E ⊂ A(n, k) with dim E = s such that B has an at least α-dimensional intersection with each k-dimensional affine subspace of the family E, that is, dim (B ∩ P ) ≥ α for all P ∈ E. What is the smallest possible value of dim B (as a function of α, s, n, k)? In [5] it was proved that if B ⊂ R n is an (α, k, s)-Furstenberg set, then dim B ≥ 2α − k + min{s, 1}. The method used in [5] generalizes the method of Wolff [12] yielding the lower bound 2α for classical plane α-Furstenberg-sets.…”

“…In [5] it was proved that if B ⊂ R n is an (α, k, s)-Furstenberg set, then dim B ≥ 2α − k + min{s, 1}. The method used in [5] generalizes the method of Wolff [12] yielding the lower bound 2α for classical plane α-Furstenberg-sets.…”

“…In [5] the authors proved the following. Theorem 1.10 (Héra-Keleti-Máthé, [5]). Let ∅ = E ⊂ A(n, k), and put B =…”

“…Remark 1.11. It is not hard to see ( [5]) that for any ∅ = E ⊂ A(n, k) we have dim P ∈E P ≤ k+dim E. This implies in particular that for any ∅ = E ⊂ A(n, k) with dim E ∈ [0, 1], dim P ∈E P = k + dim E. In this paper we also obtain new bounds for unions of affine subspaces as a corollary, namely, by applying Theorem 1.2 for B = P ∈E P ⊂ R n and α = k. Corollary 1.12. Let ∅ = E ⊂ A(n, k), and put B = P ∈E P ⊂ R n .…”