Let $\mathbb{k}$ be a Local field (then there is an integer $q=p^r$ where $p$ is a fixed prime element of $\mathbb{k}$ and $r$ is a positive integer) and
Let $\{u(n)\}_{n=0}^{+\infty}$ be a complete family of coset representatives of compact group $D=\{x\in \mathbb{k}: |x|\le1\}$ in $\mathbb{k}$ and
Set $\Gamma_{j,n}:=\{x\in \mathbb{k}: |x+u(n)| \le q^{j+1} \}$,
then
for sufficiently small $j$, the collection $\{\Gamma_{j,n} \}_{n=0}^{+\infty}$ is a disjoint collection, that is
$$\bigcap\limits_{n=0}^{+\infty} \Gamma_{j,n}=\emptyset.$$