Center of Distances and Central Cantor Sets

Abstract

We study a recently discovered metric invariant - the center of distances. The center of distances of a nonempty subset A of a metric space $$(X,\,d)$$ ( X , d ) is defined by $$S(A) :=\{ \alpha \in [0,\,+\infty ):\ \forall \ x\in A\ \ \exists \ y\in A d(x,\,y)=\alpha \} $$ S ( A ) : = { α ∈ [ 0 , + ∞ ) : ∀ x ∈ A ∃ y ∈ A d ( x , y ) = α } . Given a nonincreasing sequence $$(a_{n})$$ ( a n ) of positive numbers converging to 0, the set $$E(a_{n})\ :=\ \left\{ x\in {\mathbb {R}}:\ \exists A\subset {\mathbb {N}} \ \ x=\,\sum _{n\in A}a_{n}\right\} $$ E ( a n ) : = x ∈ R : ∃ A ⊂ N x = ∑ n ∈ A a n is called the achievement set of the sequence $$(a_{n})$$ ( a n ) . This new invariant is particularly useful in investigating achievability of sets on the real line. We concentrate on computing the centers of distances of central Cantor sets. Any central Cantor set C is an achievement set of exactly one fast convergent series $$ \sum a_{n}$$ ∑ a n , and consequently $$S(C)\supset \left\{ 0\right\} \cup \left\{ a_{n}:n\in {\mathbb {N}}\right\} $$ S ( C ) ⊃ 0 ∪ a n : n ∈ N . We try to check which central Cantor sets have the minimal possible center of distances and which have not.