Small Union with Large Set of Centers

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.