Characterization of Approximately Geometric Linear Recurrences

Characterize all triples consisting of a matrix $A\in S^{d\times d}$ and vectors $x,y\in S^d$ for which the sequence $a_n=x^TA^ny$ is approximately geometric in a preordered semiring.

Background

The paper studies sequences generated by linear recurrences over preordered semirings, represented in matrix form as $a_n=x^TA^ny$. It establishes sufficient conditions ensuring that such a sequence is approximately geometric and shows that, when $0\le 1$, primitive matrices provide an important class satisfying those conditions.

The authors explicitly state that a general characterization of the triples $(A,x,y)$ yielding approximately geometric sequences is unavailable. Such a characterization would identify precisely when the long-term behavior of a semiring-valued linear recurrence can be represented, up to asymptotic equivalence, by a geometric sequence.