In the paper

The volume of a small geodesic ball in a Riemannian manifold, Michigan Mathematical Journal 20 (1973),

A. Gray proved the following result:

Let $M$ be a Riemannian manifold of dimension $n$ with positive scalar Ricci curvature at $x \in M$. If $R$ is a sufficiently small real number, calling $B(x, \, R)$ a geodesic ball of radius $R$ centered at $x$, we have $$\mathrm{Vol} \,(B(x, \, R)) \leq C R^n,$$ where $C$ is a constant depending only on $n$.

My question involves balls of sufficiently big radius, instead of sufficiently small one. In other words, let me as what follows:

Let $M \subset \mathbb{R}^N$ be an embedded submanifold of dimension $n$. Under which conditions on $M$ and $x \in M$ it is possible to find constants $C$ and $\gamma$, depending only on $n$, such that $$\mathrm{Vol} \,(B(x, \, R)) \leq C R^{\gamma}$$ for all $R >>0$?

I am not an expert on differential geometry, so I apologize in advance if my formulation of the problem is not the optimal one, or if the answer is somehow trivial.

Any reference to the relevant literature will be greatly appreciated.