TY - JOUR

T1 - How to avoid a compact set

AU - Fornasiero, Antongiulio

AU - Hieronymi, Philipp

AU - Walsberg, Erik

N1 - Funding Information:
The second author was partially supported by NSF grants DMS-1300402 and DMS-1654725. The first and the third author were partially supported by the European Research Council under the European Union's Seventh Framework Programme (FP7/2007–2013) / ERC Grant agreement no. 291111/MODAG.
Publisher Copyright:
© 2017 Elsevier Inc.

PY - 2017/9/7

Y1 - 2017/9/7

N2 - 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.

AB - 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.

KW - Expansions of the real ordered additive group

KW - Hausdorff dimension

KW - Marstrand projection theorem

KW - Packing dimension

KW - Tame geometry

KW - Topological dimension

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U2 - 10.1016/j.aim.2017.07.011

DO - 10.1016/j.aim.2017.07.011

M3 - Article

AN - SCOPUS:85025695404

SN - 0001-8708

VL - 317

SP - 758

EP - 785

JO - Advances in Mathematics

JF - Advances in Mathematics

ER -