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Published in:
2017 | OriginalPaper | Chapter
Small Union with Large Set of Centers
Author : Tamás Keleti
Published in: Recent Developments in Fractals and Related Fields
Publisher: Springer International Publishing
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Abstract
Let \(T \subset \mathbb{R}^{n}\) be a fixed set. By a scaled copy of T around \(x \in \mathbb{R}^{n}\) we mean a set of the form x + rT for some r > 0. In this survey paper we study results about the following type of problems: How small can a set be if it contains a scaled copy of T around every point of a set of given size? We will consider the cases when T is circle or sphere centered at the origin, Cantor set in \(\mathbb{R}\), the boundary of a square centered at the origin, or more generally the k-skeleton (0 ≤ k < n) of an n-dimensional cube centered at the origin or the k-skeleton of a more general polytope of \(\mathbb{R}^{n}\). We also study the case when we allow not only scaled copies but also scaled and rotated copies and also the case when we allow only rotated copies.