A characterization of Benford's Law in discrete-time linear systems

TL;DR

This paper establishes a necessary and sufficient condition, called 'nonresonance', for solutions of linear difference equations to follow Benford's Law, unifying and extending previous results in the literature.

Contribution

It introduces the 'nonresonance' condition as a comprehensive criterion for Benford behavior in linear systems, encompassing all prior special cases.

Findings

The 'nonresonance' condition characterizes when solutions follow Benford's Law.

The condition applies to all solutions of autonomous linear difference equations.

Implications for number theory and potential extensions are discussed.

Abstract

A necessary and sufficient condition ("nonresonance") is established for every solution of an autonomous linear difference equation, or more generally for every sequence $(x^\top A^n y)$ with $x,y\in \mathbb{R}^d$ and $A\in \mathbb{R}^{d\times d}$, to be either trivial or else conform to a strong form of Benford's Law (logarithmic distribution of significands). This condition contains all pertinent results in the literature as special cases. Its number-theoretical implications are discussed in the context of specific examples, and so are its possible extensions and modifications.