Most linear flows on $\mathbb{R}^d$ are Benford

TL;DR

This paper establishes a precise condition called exponential nonresonance under which signals from linear flows on bR^d either vanish or follow Benford's Law, unifying previous results and showing this behavior is typical.

Contribution

It introduces the concept of exponential nonresonance as a necessary and sufficient condition for Benford behavior in linear flows, extending and unifying prior work.

Findings

Exponential nonresonance characterizes Benford behavior in linear flows.

Most linear flows are exponentially nonresonant, making Benford's Law typical.

The results unify previous sufficient conditions for Benford's Law in linear systems.

Abstract

A necessary and sufficient condition ("exponential nonresonance") is established for every signal obtained from a linear flow on $\mathbb{R}^d$ by means of a linear observable to either vanish identically or else exhibit a strong form of Benford's Law (logarithmic distribution of significant digits). The result extends and unifies all previously known (sufficient) conditions. Exponential nonresonance is shown to be typical for linear flows, both from a topological and a measure-theoretical point of view.