On the center of distances

TL;DR

This paper introduces the concept of the center of distances in metric spaces, extending classical theorems and analyzing specific sets like the Cantorval to deepen understanding of sequence subsums and their properties.

Contribution

It defines the center of distances for metric spaces, generalizes von Neumann's permutation theorem, and computes this center for the Cantorval and related sets.

Findings

Computed the center of distances for the Cantorval.

Extended von Neumann's theorem to compact metric spaces.

Provided new insights into sets of subsums of positive sequences.

Abstract

In this paper we introduce the notion of the center of distances of a metric space, which is required for a generalization of the theorem by J. von Neumann about permutations of two sequences with the same set of cluster points in a compact metric space. Also, the introduced notion is used to study sets of subsums of some sequences of positive reals, as well for some impossibility proofs. We compute the center of distances of the Cantorval, which is the set of subsums of the sequence $\frac34, \frac12, \frac3{16}, \frac18, \ldots , \frac3{4^n}, \frac2{4^n}, \ldots$, and also for some related subsets of the reals.