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

A necessary and sufficient condition (“nonresonance”) is established for every solution of an autonomous linear difference equation, or more generally for every sequence $(x^{\mathrm{\top }}{A}^{n}y)$ 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.