On nearly linear recurrence sequences

TL;DR

This paper introduces nearly linear recurrence sequences (nlrs), providing an asymptotic formula, comparing them with linear recurrence sequences (lrs), and exploring their properties, especially regarding finiteness and common terms, revealing significant differences from traditional lrs.

Contribution

The paper develops a Binet-type formula for nlrs, compares nlrs with associated lrs, and demonstrates key differences in their properties, especially in finiteness results and common term behavior.

Findings

Difference sequences tend to infinity under certain conditions.

Finiteness results for lrs do not hold for nlrs with integer terms.

Common terms of two nlrs are very sparse under certain hypotheses.

Abstract

A nearly linear recurrence sequence (nlrs) is a complex sequence $(a_n)$ with the property that there exist complex numbers $A_0$,$\ldots$, $A_{d-1}$ such that the sequence $\big(a_{n+d}+A_{d-1}a_{n+d-1}+\cdots +A_0a_n\big)_{n=0}^{\infty}$ is bounded. We give an asymptotic Binet-type formula for such sequences. We compare $(a_n)$ with a natural linear recurrence sequence (lrs) $(\tilde{a}_n)$ associated with it and prove under certain assumptions that the difference sequence $(a_n- \tilde{a}_n)$ tends to infinity. We show that several finiteness results for lrs, in particular the Skolem-Mahler-Lech theorem and results on common terms of two lrs, are not valid anymore for nlrs with integer terms. Our main tool in these investigations is an observation that lrs with transcendental terms may have large fluctuations, quite different from lrs with algebraic terms. On the other hand we show under certain hypotheses, that though there may be infinitely many of them, the common terms of two nlrs are very sparse. The proof of this result combines our Binet-type formula with a Baker type estimate for logarithmic forms.