Nearly subadditive sequences

TL;DR

This paper investigates the conditions under which sequences exhibit nearly subadditive behavior, establishing the necessity of the de Bruijn-Erdős condition for the error term in Fekete's Lemma and exploring the convergence of sequence slopes.

Contribution

It proves the necessity of the de Bruijn-Erdős condition for the error term in Fekete's Lemma and extends results to cases where the error term is bounded, showing convergence of sequence slopes.

Findings

Sequences with divergent error series can have slopes taking all rational values.

Bounded error terms under certain conditions ensure the existence of a limit for the sequence of slopes.

The de Bruijn-Erdős condition is both necessary and sufficient for the error term in Fekete's Lemma.

Abstract

We show that the de Bruijn-Erd\H{o}s condition for the error term in their improvement of Fekete's Lemma is not only sufficient but also necessary in the following strong sense. Suppose that given a sequence $0\leq f(1)\leq f(2)\leq f(3)\leq \dots $ such that \begin{equation}\sum_{ n=1}^{\infty} f(n)/n^2 = \infty. \end{equation} Then, there exists a sequence $\{b(n)\}_{n=1,2,\dots}$ satisfying \begin{equation}\label{eq1} b(n+m) \leq b(n) + b(m) + f(n+m) \end{equation} such that the sequence of slopes $\{ b(n)/n\}_{n=1,2,\dots}$ takes every rational number. When the series is bounded we improve their result as follows. If there exist $N$ and real $\mu >1$ such that near $f$-subadditivity holds for all pairs $(n,m)$ with $N\leq n\leq m \leq \mu n$, then $\lim_n b(n)/n $ exists.