Paradoxical sets and sets with two removable points

Abstract

A set $E \subset \mathbb {R}^n$ is called uniformly discrete if there exists an $\varepsilon >0$ such that no two points of E are closer than $\varepsilon$. Applying a theorem of T. Tao on the absence of paradoxical decompositions of uniformly discrete sets we will prove, under an additional assumption, that such a set $E \subset \mathbb {R}^n$ has at most one point p such that $E {\setminus } \{p\}$ and E are congruent. We prove also that if $E \subseteq \mathbb {R}^n$ is a discrete set and G is a discrete subgroup of the group of isometries of $\mathbb {R}^n$ then there is at most one point $p \in E$ such that there exists a $\varphi \in G$ with $\varphi (E) = E {\setminus } \{p\}$. Related unsolved problems will be pointed out.