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 [5]; 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 $k+1$.