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

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