Nearly subadditive sequences

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.