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 < α ≤ k, if B ⊂ R n and E is a nonempty collection of k-dimensional affine subspaces of R n such that every P ∈ E intersects B in a set of Hausdorff dimension at least α, then dim B ≥ 2α − k + min(dim E, 1), where dim denotes the Hausdorff dimension. As a consequence, we generalize the well-known Furstenberg-type estimate that every α-Furstenberg set has Hausdorff dimension at least 2α; we strengthen a theorem of Falconer and Mattila [5]; and we show that for any 0 ≤ k < n, if a set A ⊂ 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.

Our aim is to find the minimal Hausdorff dimension of the union of scaled and/or rotated copies of the k-skeleton of a fixed polytope centered at the points of a given set. For many of these problems, we show that a typical arrangement in the sense of Baire category gives minimal Hausdorff dimension. In particular, this proves a conjecture of R. Thornton.Our results also show that Nikodym sets are typical among all sets which contain, for every x ∈ R n , a punctured hyperplane H \ {x} through x. With similar methods we also construct a Borel subset of R n of Lebesgue measure zero containing a hyperplane at every positive distance from every point.2010 Mathematics Subject Classification. 28A78, 54E52.

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We show that if B ⊂ R n and E ⊂ A(n, k) is a nonempty collection of k-dimensional affine subspaces of R n such that every P ∈ E intersects B in a set of Hausdorff dimension at least α with k − 1 < α ≤ k, then dim B ≥ α + dim E/(k + 1), where dim denotes the Hausdorff dimension. This estimate generalizes the well known Furstenberg-type estimate that every α-Furstenberg set in the plane has Hausdorff dimension at least α + 1/2.More generally, we prove that if B and E are as above with 0 < α ≤ k, then dim B ≥ α + (dim E − (k − α )(n − k))/( α + 1). We also show that this bound is sharp for some parameters.As a consequence, we prove that for any 1 ≤ k < n, the union of any nonempty s-Hausdorff dimensional family of k-dimensional affine subspaces of R n has Hausdorff dimension at least k + s k+1 .