On the center of distances

Wojciech Bielas,Szymon Plewik,Marta Walczyńska

doi:10.1007/s40879-017-0199-4

Abstract

We introduce the notion of a center of distances of a metric space and use it in a generalization of the theorem by John von Neumann on permutations of two sequences with the same set of cluster points in a compact metric space. This notion is also 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 frac{3}{4}, frac{1}{2}, frac{3}{16}, frac{1}{8}, ldots , frac{3}{4^n}, frac{2}{4^n}, ldots , and for other related subsets of the reals.

Highlights

The center of distances seems to be an elementary and natural notion which, as far as we know, has not been studied in the literature
We introduce the notion of a center of distances of a metric space and use it in a generalization of the theorem by John von Neumann on permutations of two sequences with the same set of cluster points in a compact metric space
We compute the center of distances of the Cantorval, which is the set of subsums of the sequence

Summary

Introduction

The center of distances seems to be an elementary and natural notion which, as far as we know, has not been studied in the literature. It is an intuitive and natural concept which allows us to prove a generalization of von Neumann’s theorem on permutations of two sequences with the same set of cluster points in a compact metric space, see Theorem 2.1. We present the use of this notion for impossibility proofs, i.e., to show that a given set cannot be the set of subsums, for example see Corollary 5.5. We refer the readers to the paper [14] by Nitecki, as it provides a good introduction to facts about the set of subsums of a given sequence. Results concerning some properties of X are discussed in Propositions 4.1, 4.3 and 4.4; Corollary 4.5; Theorems 5.2, 5.3, 5.4, 6.1 and 6.2; and they are presented in Figs. 1, 2 and 3

A generalization of von Neumann’s theorem

On the center of distances and the set of subsums

An example of a Cantorval

Computing centers of distances

Digital representation of points in the Cantorval X