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 -