Let $G$ be a vertex-transitive non-ameanable graph and let $B(x,n)$ be the ball of radius $n$ centered on the vertex $x$. I am interested in estimates on the cardinality of the following set,
$$B(0,n) \cap B(x,n),$$
where $x$ is a vertex on the internal boundary of $B(0,n)$ and $1 << k << n$. I would like to show that this number is much smaller than the cardinality of $B(x,k)$. This is certainly the case for a tree. Which methods could I use to prove it on general vertex-transitive non-ameanable graphs?