Recurrence for upper Banach Density

Question

Let $E\subseteq \mathbb{Z}$ have $d^*(E) >0$ where $d^*$ is upper Banach density. I am trying to understand why there must exist an $1\leq n \leq \frac{1}{d^*(E)} + 1$ such that $d^*(E\cap (E-n)) > 0$. I see the similarity to Poincare recurrence, but $d^*$ does not seem to have the properties needed to make the analogous proof. Any help would be appreciated.