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

TL;DR

This paper establishes new lower bounds on the Hausdorff dimension of sets intersecting families of affine subspaces, generalizing Furstenberg set estimates to higher dimensions and affine subspaces.

Contribution

It introduces generalized Hausdorff dimension bounds for Furstenberg-type sets associated with affine subspaces, extending known planar results to higher dimensions.

Findings

Derived lower bounds for Hausdorff dimension of sets intersecting affine subspaces.

Proved the bounds are sharp for certain parameters.

Established that unions of families of affine subspaces have dimension at least a specific lower bound.

Abstract

We show that if $B \subset \mathbb{R}^n$ and $E \subset A(n,k)$ is a nonempty collection of $k$-dimensional affine subspaces of $\mathbb{R}^n$ such that every $P \in E$ intersects $B$ in a set of Hausdorff dimension at least $\alpha$ with $k-1 < \alpha \leq k$, then $\dim B \geq \alpha +\dim E/(k+1)$, where $\dim$ denotes the Hausdorff dimension. This estimate generalizes the well known Furstenberg-type estimate that every $\alpha$-Furstenberg set in the plane has Hausdorff dimension at least $\alpha + 1/2$. More generally, we prove that if $B$ and $E$ are as above with $0 < \alpha \leq k$, then $\dim B \geq \alpha +(\dim E-(k-\lceil \alpha \rceil)(n-k))/(\lceil \alpha \rceil+1)$. We also show that this bound is sharp for some parameters. As a consequence, we prove that for any $1 \leq k<n$, the union of any nonempty $s$-Hausdorff dimensional family of $k$-dimensional affine subspaces of $\mathbb{R}^n$ has Hausdorff dimension at least $k+\frac{s}{k+1}$.