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 $k-1 < \alpha \leq k$, 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

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与仿射子空间族相关的furstenberg型集的Hausdorff维数

我们证明了如果$B \subset \mathbb{R}^n$和$E \subset A(n,k)$是$\mathbb{R}^n$的$k$的仿射子空间的非空集合，使得每一个$P \in E$都与$B$在至少$\alpha$与$k-1 < \alpha \leq k$的Hausdorff维数集合中相交，则$\dim B \geq \alpha +\dim E/(k+1)$，其中$\dim$表示Hausdorff维数。这个估计推广了众所周知的furstenberg型估计，即平面上的每个$\alpha$ -Furstenberg集至少具有$\alpha + 1/2$的Hausdorff维数。更一般地，我们证明如果$B$和$E$与$0 < \alpha \leq k$相同，则$\dim B \geq \alpha +(\dim E-(k-\lceil \alpha \rceil)(n-k))/(\lceil \alpha \rceil+1)$。我们还证明了这个界对于某些参数是尖锐的。由此证明了对于任意$1 \leq k

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