Small unions of affine subspaces and skeletons via Baire category

Chang, A. and Csörnyei, M. and Héra, Kornélia and Keleti, Tamás (2018) Small unions of affine subspaces and skeletons via Baire category. ADVANCES IN MATHEMATICS, 328. pp. 801-821. ISSN 0001-8708

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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. © 2018

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