Small union with large set of centers

TL;DR

This survey explores the minimal size of sets that contain scaled, rotated, or both scaled and rotated copies of various geometric objects around every point of a given set, covering classical and generalized shapes.

Contribution

It provides a comprehensive overview of results on the minimal size of sets containing scaled and rotated copies of geometric structures, including new insights into various configurations.

Findings

Sets containing scaled copies of circles or spheres can be very small.

Allowing rotations changes the minimal size requirements for such sets.

Results vary depending on the shape and the type of copies (scaled, rotated, or both).

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.