Small unions of affine subspaces and skeletons via Baire category

TL;DR

This paper investigates the minimal Hausdorff dimension of unions of scaled or rotated skeletons of polytopes and shows that typical arrangements, in the sense of Baire category, achieve this minimal dimension, confirming a conjecture and exploring properties of Nikodym sets.

Contribution

It proves that typical arrangements of scaled and rotated skeletons of polytopes have minimal Hausdorff dimension, confirming R. Thornton's conjecture, and explores properties of Nikodym sets and measure-zero sets containing hyperplanes.

Findings

Typical arrangements achieve minimal Hausdorff dimension.

Confirmed a conjecture of R. Thornton.

Constructed measure-zero sets containing hyperplanes at every positive distance.

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.