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 ($0\le k<n$) 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.