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

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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.

Keywords

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

Mathematics Subject Classification

37A05 37A45 11J71 62E20 

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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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