Center of distances and Bernstein sets

TL;DR

This paper demonstrates that for any subset of non-negative real numbers containing zero, there exists a Bernstein set whose set of distances between points matches that subset.

Contribution

It establishes a general construction linking subsets of non-negative reals to Bernstein sets with prescribed distance sets.

Findings

Existence of Bernstein sets with prescribed distance sets for any subset containing zero.

Generalization of the concept of distance sets in the context of Bernstein sets.

Provides a method to realize any such subset as a distance set of a Bernstein set.

Abstract

We show that for any subset $A\subset [0,\infty)$, where $0\in A$, there exists a Bernstein set $X\subset \mathbb R$ such that $A$ is the center of distances of $X$.