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

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 $k-1<\alpha \le k$, then $\dim B\ge \alpha +\dim E/(k+1)$, 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 $0<\alpha \le k$, then $\dim B\ge \alpha +(\dim E-(k-\lceil \alpha \rceil )(n-k))/(\lceil \alpha \rceil +1)$. We also show that this bound is sharp for some parameters. As a consequence, we prove that for any $1\le k<n$, the union of any nonempty $s$-Hausdorff dimensional family of $k$-dimensional affine subspaces of ${\mathbb{R}}^n$ has Hausdorff dimension at least $k+\frac{s}{k+1}$.