Paradoxical sets and sets with two removable points

A set E⊂ Rn is called uniformly discrete if there exists an ε> 0 such that no two points of E are closer than ε. 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⊂ Rn has at most one point p such that E { p} and E are congruent. We prove also that if E⊆ Rn is a discrete set and G is a discrete subgroup of the group of isometries of Rn then there is at most one point p∈ E such that there exists a φ∈ G with φ(E) = E { p}. Related unsolved problems will be pointed out. © 2018, Springer International Publishing AG, part of Springer Nature.