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

We prove that for any $1 \le k<n$ and $s\le 1$, the union of any nonempty $s$-Hausdorff dimensional family of $k$-dimensional affine subspaces of ${\mathbb R}^n$ has Hausdorff dimension $k+s$. More generally, we show that for any $0 < \alpha \le k$, if $B \subset {\mathbb R}^n$ and $E$ 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 $\alpha$, then $\dim B \ge 2 \alpha - k + \min(\dim E, 1)$, where $\dim$ denotes the Hausdorff dimension. As a consequence, we generalize the well known Furstenberg-type estimate that every $\alpha$-Furstenberg set has Hausdorff dimension at least $2 \alpha$; we strengthen a theorem of Falconer and Mattila; and we show that for any $0 \le k<n$, if a set $A \subset {\mathbb R}^n$ contains the $k$-skeleton of a rotated unit cube around every point of ${\mathbb R}^n$, or if $A$ contains a $k$-dimensional affine subspace at a fixed positive distance from every point of ${\mathbb R}^n$, then the Hausdorff dimension of $A$ is at least $k + 1$.