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

We prove that for any 1 <= k < n and s <= 1, the union of any nonempty s-Hausdorff dimensional family of k-dimensional affine subspaces of R-n has Hausdorff dimension k + s. More generally, we show that for any 0 < alpha <= k, if B subset of R-n and E is a nonempty collection of k-dimensional affine subspaces of R-n such that every P is an element of E intersects B in a set of Hausdorff dimension at least alpha, then dim B >= 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 <= k < n, if a set A subset of R-n contains the k-skeleton of a rotated unit cube around every point of R-n, or if A contains a k-dimensional affine subspace at a fixed positive distance from every point of R-n, then the Hausdorff dimension of A is at least k + 1.