Small unions of affine subspaces and skeletons via Baire category

Abstract

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 point x of 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.

to appear in Advances in Mathematics

References in corpus (2)

• Cubes and Their Centers
• Hausdorff dimension of unions of affine subspaces and of Furstenberg-type sets

Cited by in corpus (7)

• Cubes and Their Centers
• The Kakeya needle problem and the existence of Besicovitch and Nikodym sets for rectifiable sets
• Hausdorff dimension of unions of affine subspaces and of Furstenberg-type sets
• Large Sets with Small Injective Projections
• Small union with large set of centers
• A strong-type Furstenberg-Sárközy theorem for sets of positive measure
• Pattern Problems related to the Arithmetic Kakeya Conjecture