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

  • Arno Berger
  • Gideon Eshun


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 



The authors have been supported by an Nserc Discovery Grant. They like to thank T.P. Hill, B. Schmuland, M. Waldschmidt, A. Weiss and R. Zweimüller for helpful discussions and comments.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Mathematical and Statistical SciencesUniversity of AlbertaEdmontonCanada

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