A Scale-Invariant Entropy Statistic for Distance Distributions

TL;DR

This paper introduces a new family of scale-invariant entropy statistics based on logarithmic aggregation of distance distributions, capturing structural features of point configurations regardless of scale.

Contribution

The work presents a novel methodological approach for deriving scale-invariant entropy measures from distance distributions, applicable to point processes.

Findings

The entropy statistics are insensitive to absolute scale changes.

The method encodes structural features of point configurations.

Applicable to various configurations, including prime number-based examples.

Abstract

We introduce a family of scale-invariant entropy statistics derived from logarithmically aggregated distance distributions of point processes, with prime numbers serving as a motivating example. The construction associates to each finite configuration a scalar quantity encoding structural features of relative spacing while remaining insensitive to absolute scale. This work is intended as a methodological contribution rather than a source of new raw results.