Completeness of Positive Linear Recurrence Sequences

TL;DR

This paper classifies the completeness of positive linear recurrence sequences (PLRS) using Brown's criterion, providing characterizations for various families and proposing bounds for the completeness region based on the characteristic polynomial's root.

Contribution

It offers a complete classification of PLRS completeness for several families and introduces a more efficient method using root bounds to determine completeness.

Findings

Complete classification for several PLRS families.

A new method to check completeness via characteristic polynomial roots.

Identification of an indeterminate region where root size does not determine completeness.

Abstract

A sequence of positive integers is complete if every positive integer is a sum of distinct terms. A positive linear recurrence sequence (PLRS) is a sequence defined by a homogeneous linear recurrence relation with nonnegative coefficients of the form $H_{n+1} = c_1 H_n + \cdots + c_L H_{n-L+1}$ and a particular set of initial conditions. We seek to classify various PLRS's by completeness. With results on how completeness is affected by modifying the recurrence coefficients of a PLRS, we completely characterize completeness of several families of PLRS's as well as conjecturing criteria for more general families. Our primary method is applying Brown's criterion, which says that an increasing sequence $\{H_n\}_{n = 1}^{\infty}$ is complete if and only if $H_1 = 1$ and $H_{n + 1} \leq 1 + \sum_{i = 1}^n H_i$. %A survey of these results can be found in \cite{BHLLMT}. Finally, we adopt previous analytic work on PLRS's to find a more efficient way to check completeness. Specifically, the characteristic polynomial of any PLRS has exactly one positive root; by bounding the size of this root, the majority of sequences may be classified as complete or incomplete. Additionally, we show there exists an indeterminate region where the principal root does not reveal any information on completeness. We have conjectured precise bounds for this region.