Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

19
The categories of this paper are: eess
16
The categories of this paper are: stat
24
The categories of this paper are: dg-ga
5
The categories of this paper are: gr-qc
8
The categories of this paper are: hep-ex
25
The categories of this paper are: patt-sol
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics
17
The categories of this paper are: q-fin
22
The categories of this paper are: chao-dyn
18
The categories of this paper are: econ
36
The categories of this paper are: cmp-lg
21
The categories of this paper are: adap-org
23
The categories of this paper are: solv-int
28
The categories of this paper are: alg-geom
9
The categories of this paper are: nlin