Journal of Dynamics and Differential Equations

, Volume 28, Issue 2, pp 431–469

A Characterization of Benford’s Law in Discrete-Time Linear Systems


DOI: 10.1007/s10884-014-9393-y

Cite this article as:
Berger, A. & Eshun, G. J Dyn Diff Equat (2016) 28: 431. doi:10.1007/s10884-014-9393-y


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.


Benford sequence Uniform distribution mod 1 \(\mathbb {Q}\)-independence Nonresonant set 

Mathematics Subject Classification

37A05 37A45 11J71 62E20 

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada