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.
2026-04-12 09:56:32.1775987792
Translation of a set
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No.
Take for instance the Cantor set on $[0,1]$
We have that $m(C)=0$ and $C-C=[-1,1]$
Thus $(0,1) \subseteq C-C\Longrightarrow C \cap(C+r) \neq \emptyset,\forall r \in (0,1)$