Subadditivity question I learned Fekete's subadditivity lemma for sequences, so I would like to know some  interesting examples of arrays that are subadditive?
Any hint is welcome, thanks in advance.
 A: Take any subadditive function $f:[0,\infty)\rightarrow[0,\infty)$ and define $a_n:=f(n)$.
If by "array" you mean a double indexed sequence for which
$$a_{k+m,l+n}\leq a_{k,l}+a_{m,n}$$
You can always take two subadditive sequences $a_n,b_m$ and define $c_{n,m}=a_n+b_m$ or $c_{n,m}=a_{n+m}$. More generally, you can take any subadditive function $f:[0,\infty)^2\rightarrow[0,\infty)$ and define $a_{m,n}:=f(m,n)$. For example, any supremum of linear functions is sub-additive so
$$a_{m,n}=\sup_{(\alpha,\beta)\in A}\{\alpha m+\beta n\}$$
where $A$ is some subset of $\mathbb{R}^2$.
A: There is, IMHO, a very elegant application of Fekete's Lemma in fixed point theory. It allows to generalize the Banach fixed point theorem with assumption on the map which is invariant of the metric up to equivalence.

All the details of the following discussion can be found in the Section 3.2. of
Khamsi, M. A., & Kirk, W. A. (2011). An introduction to metric spaces and fixed point theory (Vol. 53). John Wiley & Sons.
For a general discussion on subadditive functions ("generalizations" of Fekete's Lemma) I recommend chapter VII of Hille, E., & Phillips, R. S. (1996). Functional analysis and semi-groups (Vol. 31). American Mathematical Soc..

Let $(M,d)$ be a complete metric space and $f\colon M \to M$. $x\in M$ is a fixed point of $f$ if $f(x)=x$. Define the smallest Lipschitz constant of $f$ with respect to $d$ as
$$Lip(f,d) = \inf\{\alpha \mid d(f(x),f(y))\leq \alpha \,d(x,y),\forall x,y\in M\}.$$
Banach fixed point theorem says that

Theorem 1. If $Lip(f,d)<1$, then $f$ has a unique fixed point in $M$.

Now, for every $k\geq 1$, let $f^k = f\circ \ldots \circ f$ be the $k$-the composition of $f$ with itself. 
The following result generalizes the Banach fixed point theorem stated above.

Theorem 2. If $\displaystyle\inf_{k\geq 1}Lip(f^k,d)^{1/k}<1$, then $f$ has a unique fixed point in $M$.

Fekete's Lemma help a lot to understand the real improvement offered by Theorem 2 over Theorem 1.
Consider the sequence
$$a_k = \ln(Lip(f^k,d))\qquad \forall k\geq 1.$$
Then, $(a_k)_k$ is subadditive since, by the property of Lipschitz constants, for $k,m\geq 1$, we have
\begin{align*}
a_{k+m} &= \ln(Lip(f^{k+m},d))\\
&\leq\ln(Lip(f^k,d)Lip(f^m,d))\\
&\leq \ln(Lip(f^k,d))+\ln(Lip(f^m,d))=a_k+a_m.
\end{align*}
Fekete's Lemma now implies that
$$\inf_{k\geq 1}\frac{a_k}{k} = \lim_{k\to \infty }\frac{a_k}{k}.$$
Since the logarithm is continuous, bijective and monotonic, we have 
$$\inf_{k\geq 1}\frac{a_k}{k}= \inf_{k\geq 1}\frac{\ln(Lip(f^k,d))}{k}=\ln\Big(\inf_{k\geq 1}Lip(f^k,d)^{1/k}\Big).
$$
With a similar observation for $\lim_{k\to \infty }\frac{a_k}{k}$, we conclude that 
$$\inf_{k\geq 1}Lip(f^k,d)^{1/k}=\lim_{k\to \infty } Lip(f^k,d)^{1/k}$$
Suppose now that $d'$ is another metric on $M$ for which there exists $0<\alpha,\beta$ such that 
$$ \alpha\, d(x,y) \leq  d'(x,y) \leq \beta \, d(x,y)\qquad \forall x,y\in M.\tag{$*$}$$
We show using Fekete's Lemma that for every metric $d'$ as above, it holds $$\displaystyle\inf_{k\geq 1 }Lip(f^k,d)^{1/k}=\inf_{k\geq 1 }Lip(f^k,d')^{1/k}.$$ 
This is a quite fundamental observation as it implies that the assumption $\displaystyle\inf_{k\geq 1}Lip(f^k,d)^{1/k}<1$ in Theorem 2 only depends on $d$ up to equivalence. This is a good thing since the fact of having a fixed point only depends on $f$.
It holds 
$$\frac{\alpha}{\beta}Lip(f^k,d)\leq Lip(f^k,d') \leq \frac{\beta}{\alpha}Lip(f^k,d)\qquad \forall k \geq 1,$$
and thus 
$$\lim_{k\to \infty }\frac{\ln(Lip(f^k,d))}{k}=\lim_{k\to \infty }\frac{\ln(Lip(f^k,d'))}{k}.$$
With Fekete's Lemma we conclude that 
$$\inf_{k\geq 1 }Lip(f^k,d)^{1/k}=\inf_{k\geq 1 }Lip(f^k,d')^{1/k}$$
for every metric $d'$ satisfying $(*)$.
