We wonder for two "large" sets $I,J\subseteq \omega$, if $(J-J)\cap I=\emptyset$, then it must be due to certain periodic pattern.
We say $I\subseteq \omega$ is $t$-syndetic iff for every $n\in\omega$, $n+\hat{t} \in I$ for some $\hat t\leq t$.
Question: Let $t\in\omega$ and $I,J\subseteq \omega$ be $t$-syndetic so that $(J-J)\cap I=\emptyset$. Must there be an $a\in\omega$ and $A_I,A_J\subseteq a$ such that $J\subseteq^* a\omega+ A_J, I\subseteq^* a\omega+ A_I $ and $((A_J-A_J)+a\omega) \cap (A_I+a\omega) = \emptyset $
