Biggest $(-t,t)\subset A-A$ if $λ(Α)>0$ I am trying to solve a problem which  states that if $λ(E)>1$ ($λ$ being the Lebesgue measure in $\mathbb{R}$) then there exists $x,y \in E$ such that $x\neq y $ and $ x-y \in \mathbb{Z}$.
From the proof of the Steinhaus Lemma we can see that  $λ(Ε)>0\Rightarrow (-\frac{λ(Ε)}{2},\frac{λ(Ε)}{2})\in E-E$ and so it's trivial to solve the excerise.
I was wondering if we can improve the size of the interval contained in the difference set or is this the best we can hope for in general. 

It seems clear that the result you state is sharp. Let $E=(0,1)$; then $\lambda(E)=1$ but there do not exist $x,y\in E$ with $x\ne y$ and $x-y\in\Bbb Z$.
