Translation of a set Let $A\subset \mathbb R$ be Lebesgue measurable set. Is it true that if $\ \forall r\in(0,1)$
$$A\cap (A+r)\neq \emptyset$$
then $\lambda(A)>0$?
$$$$
I think that is linked to the Vitali set, but I did't manage to prove it.

No.
Take for instance the Cantor set on $[0,1]$
We have that $m(C)=0$ and  $C-C=[-1,1]$
Thus $(0,1) \subseteq C-C\Longrightarrow C \cap(C+r) \neq \emptyset,\forall r \in (0,1)$
