### Abstract

A first-order expansion of the R-vector space structure on R does not define every compact subset of every R^{n} if and only if topological and Hausdorff dimension coincide on all closed definable sets. Equivalently, if A⊆R^{k} is closed and the Hausdorff dimension of A exceeds the topological dimension of A, then every compact subset of every R^{n} can be constructed from A using finitely many boolean operations, cartesian products, and linear operations. The same statement fails when Hausdorff dimension is replaced by packing dimension.

Original language | English (US) |
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Pages (from-to) | 758-785 |

Number of pages | 28 |

Journal | Advances in Mathematics |

Volume | 317 |

DOIs | |

State | Published - Sep 7 2017 |

### Keywords

- Expansions of the real ordered additive group
- Hausdorff dimension
- Marstrand projection theorem
- Packing dimension
- Tame geometry
- Topological dimension

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Fornasiero, A., Hieronymi, P., & Walsberg, E. (2017). How to avoid a compact set.

*Advances in Mathematics*,*317*, 758-785. https://doi.org/10.1016/j.aim.2017.07.011