How to avoid a compact set

Antongiulio Fornasiero; Philipp Hieronymi; Erik Walsberg

doi:10.1016/j.aim.2017.07.011

How to avoid a compact set

Antongiulio Fornasiero, Philipp Hieronymi, Erik Walsberg

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

A first-order expansion of the R-vector space structure on R does not define every compact subset of every Rⁿ 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 Rⁿ 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)
Pages (from-to)	758-785
Number of pages	28
Journal	Advances in Mathematics
Volume	317
DOIs	https://doi.org/10.1016/j.aim.2017.07.011
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

General Mathematics

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10.1016/j.aim.2017.07.011

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abstract = "A first-order expansion of the R-vector space structure on R does not define every compact subset of every Rn if and only if topological and Hausdorff dimension coincide on all closed definable sets. Equivalently, if A⊆Rk is closed and the Hausdorff dimension of A exceeds the topological dimension of A, then every compact subset of every Rn 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.",

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KW - Packing dimension

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How to avoid a compact set

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Keywords

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