For $S\subset \mathbb{N}$ define the upper density as $D^{\ast }(S)=\limsup_{n\rightarrow \infty }\frac{\left\vert S\cap \{1,2,\ldots,n\} \right\vert }{% \left\vert n\right\vert }.$

Question: Suppose $D^{\ast }(S)>0$. Is there $n\in\mathbb{N}$ such that $(S-n )\cap S$ has positive upper density?

$S-n=\{s-n| s\in S\}$