lebesgue measurable problem

Question

Let $E$ be a Lebesgue measurable subset of $[0,1]$, and suppose that $m(E)>3/4$. Prove that $(-1/2,1/2) \subseteq E-E$. We use $E-E$ to denote the set $\{x-y:x,y \subseteq E\}$

Answer 1

Hint :Suppose that for some $\frac{p}{q} \in (-1/2, 1/2)$ does not exist any $x$ and $y$ in $E$ such that $x - y = \frac{p}{q}$, then $(E + \frac{p}{q}) \cap E = \varnothing $, therefore …

Now, what can you tell about the measure of these two disjoint sets? What's the contradiction?

Answer 2

Let $x\in[0,\frac12)$, then $E, E+x\subseteq[0,\frac32)$ and $$m(E\cap(E+x))=m(E)+m(E+x)-m(E\cup(E+x))>\frac34+\frac34-\frac32=0$$ so the sets intersect and $e=e'+x\Longrightarrow x=e-e'$.

So $[0,\frac12)\subseteq E-E$ and by symmetry also $(-\frac12,0]\subseteq E-E$.