Small unions of affine subspaces and skeletons via Baire category

Alan Chang, Marianna Csörnyei, Kornélia Héra, Tamás Keleti

Research output: Contribution to journalArticlepeer-review

3 Scopus citations

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∈Rn, a punctured hyperplane H∖{x} through x. With similar methods we also construct a Borel subset of Rn of Lebesgue measure zero containing a hyperplane at every positive distance from every point.

Original languageEnglish (US)
Pages (from-to)801-821
Number of pages21
JournalAdvances in Mathematics
Volume328
DOIs
StatePublished - Apr 13 2018
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

  • Baire category
  • Hausdorff dimension

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