**Definition:** Let $G$ be a group. For $g\in G$ and a subset $F\subseteq G$ fix
the notation $gF:=\{gf\mid f\in G\}$. A sequence $(F\_{i})\_{i\in\mathbb{N}}\subseteq G$
is called a *Følner sequence* if
\begin{eqnarray}
\nonumber
\lim\_{i\rightarrow\infty}\frac{\#(gF\_{i}\triangle F\_{i})}{\#F\_{i}}=0
\end{eqnarray}
for every $g\in G$. Here $gF\_{i}\triangle F\_{i}:=(gF\_{i}\cup F\_{i})\setminus(gF\_{i}\cap F\_{i})$
denotes the *symmetric difference* of $gF\_{i}$ and $F\_{i}$.


**Question:** Consider the integers $\mathbb{Z}$ and define $f: \mathbb{Z} \rightarrow \mathbb{R}\_{+}$
via $f(x):=\sqrt{|x|}$. Can we find a Følner sequence $(F\_{i})\_{i\in\mathbb{N}}$
such that the following holds:
\begin{eqnarray}
\nonumber
\lim\_{i\rightarrow\infty}\frac{\sum\_{k \in F\_i} [f(k)-f(k-1)]}{\#F\_{i}}>0\:?
\end{eqnarray}



Følner sequences of the integers
No. We have an estimate of the form $|f(k) - f(k-1)| \le C / \sqrt{|k|+1}$. Therefore $|f(k) - f(k-1)| \le \epsilon$ except at $O(\epsilon^{-2})$ values, so $\sum\_{k \in F\_i} |f(k) - f(k-1)| / |F\_i| \le O(\epsilon) + O(\epsilon^{-2} / |F\_i|)$. Putting $\epsilon = 1/|F\_i|^{1/3}$, we get $\sum\_{k \in F\_i} |f(k) - f(k-1)| / |F\_i| \le O(1/|F\_i|^{1/3})$. Note the Følner property is not required.


