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 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.
Original language | English (US) |
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Pages (from-to) | 801-821 |
Number of pages | 21 |
Journal | Advances in Mathematics |
Volume | 328 |
DOIs | |
State | Published - Apr 13 2018 |
All Science Journal Classification (ASJC) codes
- General Mathematics
Keywords
- Baire category
- Hausdorff dimension