The authors study extensions of M. Fekete’s Lemma [Math. Z. 17, 228–249 (1923; JFM 49.0047.01)]:
Fekete’s Lemma. Given a sequence \(0\leq f(1)\leq f(2)\leq f(3)\leq \ldots\) such that \[\sum_{n=1}^{\infty}\,\frac{f(n)}{n^2}=\infty,\] there exists a sequence \(\{b(n)\}_{n\in\mathbb{N}}\) with \[b(n+m)\leq b(n)+b(m)+f(n+m),\] such that the sequence of slopes \(\{b(n)/n\}_{n\in\mathbb{N}}\) takes every rational number as value.
The starting point of the paper is a theorem by N. G. de Bruijn and P. Erdős [Nederl. Akad. Wet., Proc., Ser. A 55 (Indag. Math. 14), 152–163 (1952; Zbl 0047.06303)] and the main results of the paper are the following.
Theorem 2. Suppose \(\mu >1\) and \(N\geq 1\) are given. If \(\{a(1),a(2),\ldots\}\) is \((\mu,N)\)-subadditive, i.e. \[a(n+m)\leq a(n)+a(m),\quad n\leq m\leq \mu n;\ n,m\geq N,\] then \(\lim_{n\rightarrow\infty}a(n)/n\) exists and equals \(\inf_{k\geq N} a(k)/k\). (It may be \(-\infty\).)
Theorem 3. Suppose \(\mu>1\) and \(N\geq 1\) are given and \(f\) is a non-negative monotone increasing real function satisfying \[\sum_{n=1}^{\infty}\,\frac{f(n)}{n^2}\text{ is finite}.\] If the sequence \(\{a(1),a(2),\ldots\}\) is \((\mu,N,f)\)-subadditive, i.e., \[a(n+m)\leq a(n)+a(m)+f(n+m),\quad m\leq n\leq \mu m; \ m,n\geq N,\] then \(\lim_{n\rightarrow\infty} a(n)/n\) exists. (It may be \(-\infty\).)
Theorem 4. Let \(f(n)\) be a non-negative, non-decreasing sequence and suppose \[\sum_{1\leq n <\infty}\,\frac{f(n)}{n^2}=\infty.\] Then there exists a nearly \(f\)-subadditive sequence \(b(1),b(2),b(3),\ldots\) of rational numbers, i.e., for all integers \(m,n\geq 1\), \[b(n+m)\leq b(n)+b(m)+f(n+m),\] such that the set of slopes takes all rationals exactly once.