Small unions of affine subspaces and skeletons via Baire category

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∈Rⁿ, a punctured hyperplane H∖{x} through x. With similar methods we also construct a Borel subset of Rⁿ of Lebesgue measure zero containing a hyperplane at every positive distance from every point.