Translation of a finite set and lebesgue measure

Question

Let $A \subset \mathbb{R}$ a finite set and $E \subset \mathbb{R} $ a lebesgue measurable set and $m(E)>0$.Prove that $\exists x\in \mathbb{R}$ and $\exists s>0$ such that $x+sA \subset E$.

I tried to use fubini's theorem and the steinhauss theorem without success.

Can someone help me with this?

Answer 1

See Theorem 3 here. The main ingredient is the Lebesgue density theorem, which you should attempt to use for this problem if you have not already.