Let $E \in [0,1]$ be a Lebesgue measurable set with positive measure. Show that there exists $a,b$ such that all three numbers $a$, $a+b$, $a+2b$ are contained in $E$.
I know that if no such $a,b$, then $E$ must contain no interval, but I know there are some Cantor type set with positive measure but it doesn’t contain any interval.
By considering $E \cap (-n,n)$ we may suppose that $E$ is bounded. Let $0 <\epsilon <m(E)$. There exists a continuous function $f$ with compact support such that $0\leq f \leq 1$ and $\int |f-I_E| <\epsilon$. Claim: there exists $n$ such that $\int f(x)f(x+\frac 1 n)f(x+\frac 2 n) >0$. To prove this claim assume that the integral is $0$ for all $n$ and let $n \to \infty$ to get $\int f^{3}=0$ which implies $f\equiv 0$. But then $\int |0-I_E| <\epsilon$, a contradiction to our choice of $\epsilon$. This proves our claim. I leave it to you to now conclude that $\int I_E I_{E-\frac 1 n}I_{E-\frac 2 n} >0$ provided $\epsilon$ is chosen appropriately. [Use triangle inequality and boundedness of $f$]. Finally this implies that $I_E(a) I_{E-\frac 1 n}(a)I_{E-\frac 2 n}(a) >0$ for some $a$. Take $b=\frac 1 n$ to finish the proof.