TITLE: If $E$ has positive measure, then prove that there exists $h \in \mathbb{R}$ such that $|(E+h) \cap E| > 0$.
QUESTION [5 upvotes]: If $E$ has positive measure, then prove that there exists $h \in \mathbb{R}$ such that $|(E+h) \cap E| > 0$. I tried doing this by contradiction but cannot seem to find my way about the argument.
Any hints or help would be appreciated. Thanks.

REPLY [2 votes]: It suffices to prove the statement for bounded $E$. So suppose $E$ is bounded.
Note $\int\int 1_E(y)1_E(y-h)dydh = \int 1_E(y)\int 1_E(y-h)dhdy = \int 1_E(y)|E|dy = |E|^2 > 0$.
So, there exists $h$ with $\int 1_E(y) 1_E(y-h)dy > 0$. But this integral is $|E\cap (h+E)|$.