Riesz energies and the magnitude of manifolds

TL;DR

This paper explores the geometric meaning of Leinster's magnitude invariant for manifolds, linking it to classical invariants and beta functions, and extends the analysis to p-adic integers using pseudodifferential techniques.

Contribution

It establishes a precise relation between magnitude invariants and classical geometric invariants for manifolds, and introduces new methods to analyze these invariants via beta functions and pseudodifferential analysis.

Findings

Relation between magnitude and Brylinski's beta function for closed manifolds

Residues of the beta function encode geometric information

Extension of the framework to p-adic integers

Abstract

We study the geometric significance of Leinster's magnitude invariant. For closed manifolds we find a precise relation with Brylinski's beta function and therefore with classical invariants of knots and submanifolds. In the special case of compact homogeneous spaces we obtain an elementary proof that the residues of the beta function contain the same geometric information as the asymptotic expansion of the magnitude function. For general closed manifolds we use the recent pseudodifferential analysis of the magnitude operator to relate these via an interpolating polynomial family. Beyond manifolds, the relation with the Brylinski beta function allows to deduce unexpected properties of the magnitude function for the $p$-adic integers.