Characterizing Benford's Law in Linear Systems

Abstract

We study the widespread logarithmic distribution of first significant digits and significands of data sets (referred to as Benford’s Law ) in the context of dynamical systems. Using recent tools and conditions under which a recursively defined sequence is Benford via the classical theory of uniform distribution modulo one, this study derives a necessary and sufficient condition (“nonresonant spectrum”) on A ∈ R d×d for every sequence (y ⊤ A n x) n∈N , with arbitrary x, y ∈ R d , emanating from the difference equation x n = Ax n−1 , to be Benford or terminating. This result in turn is used to also show that the function t → y ⊤ e tA x arising from the differential equation x(t)= Ax(t) is either Benford or identically zero for t ≥ 0. The results generalize and unify already known facts for one- and higher-dimensional systems.