## 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∈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.

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 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)

## Keywords

- Baire category
- Hausdorff dimension