Let $E\subset \mathbb{R}$ be a set of positive Lebesgue measure. Can we find $l>0$ such that $$\bigcap_{-l\leq t \leq l}t+E$$ is a set of positive Lebesgue measure?
Notation: $t+E=\{t+e|e\in E\}$
Let $E\subset \mathbb{R}$ be a set of positive Lebesgue measure. Can we find $l>0$ such that $$\bigcap_{-l\leq t \leq l}t+E$$ is a set of positive Lebesgue measure?
Notation: $t+E=\{t+e|e\in E\}$
No. Every set $E$ without interior points (e.g. the complements of the rationals) has the property that $$\bigcap_{|t|<\varepsilon}(t+E)=\emptyset$$ for every $\varepsilon>0$. Indeed, for every $x\in E$, there is $t$ with $|t|<\varepsilon$ and $x-t\notin E$, hence $x\notin\bigcap_{|t|<\varepsilon}(t+E)$.