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arXiv · 1701.01405

Small unions of affine subspaces and skeletons via Baire category

Alan Chang·Marianna Csörnyei·Kornélia Héra·Tamás Keleti
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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 point x of R^n, a punctured hyperplane H\{x} through x. With similar methods we also construct a Borel subset of R^n of Lebesgue measure zero containing a hyperplane at every positive distance from every point.

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Alan Chang, Marianna Csörnyei, Kornélia Héra, Tamás Keleti. 2017-01-05. Small unions of affine subspaces and skeletons via Baire category. https://doi.org/10.1016/j.aim.2018.02.009

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