Let $G$ be a finitely-generated group of polynomial growth equipped with the word metric (with respect to a fixed symmetric generating set).
Let \begin{equation*} A = \left\{ g \in G: |g| = mn, n \in \mathbb{N} \right\} \end{equation*} for some fixed integer $m \geq 2$.
$A$ is therefore a nested sequence of spheres in $G$.
Is there a constant $c$, perhaps depending on the growth function but independent of $g$, so that \begin{equation*} c\frac{ \#\left(B_k \cap A \right)}{\#B_k} \geq \frac{ \#\left(gB_k \cap A \right)}{\#B_k}, \end{equation*} where $B_k$ is the ball of radius $k$?
