Magnitude function determines generic finite metric spaces

TL;DR

This paper establishes conditions under which the magnitude function uniquely determines a finite metric space, especially highlighting the role of rational independence of distances in the asymptotic behavior.

Contribution

It provides new sufficient conditions for finite metric spaces to be uniquely identified by their magnitude function, emphasizing the importance of rational independence.

Findings

Finite metric spaces with rationally independent distances are determined by the magnitude function.

Asymptotic behavior of the magnitude function can uniquely identify certain finite metric spaces.

The paper offers criteria for when the magnitude function fully characterizes a finite metric space.

Abstract

We give sufficient conditions for a finite metric space to be determined by the magnitude function. In particular, a generic finite metric space such that the distances between the points are rationally independent is determined by the asymptotic behavior of the magnitude function.