Let $(M^n,g)$ be a complete flat Riemannian manifold. Suppose there exists a number $s \in (n-1,n]$ such that for some point $p \in M$
$$
\limsup\_{r \to +\infty} \frac{\text{Vol}\,B(p,r)}{r^s}>0.
$$ 
Can we prove that $(M^n,g)$ is isometric the Euclidean space?



Volume ratio of complete flat manifolds
Yes, it is true.


Note that $M$ is isometric to a quotient of the Euclidean space $\mathbb{E}^n$ by a totally discontinuous free isometric action of a group $\Gamma$.


Your condition implies that [the soul](https://en.wikipedia.org/wiki/Soul_theorem) of $M$ is a single point.
It follows that $\Gamma$ fixes a point in $\mathbb{E}^n$.
Since the action is free, the group $\Gamma$ is trivial.


