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.
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 .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.