For $E \subseteq [0, 1]$, $m(E) > 0$, show that there is $\alpha \in E$ and $\beta \neq 0$ such that $\alpha, \alpha + \beta, \alpha + 2\beta \in E$.

This was originally a proof verification question, but I have since moved the proof to an answer as discussed on meta. I still welcome comments on the proof as well as any alternative proofs.

Let $E$ be a Lebesgue measurable subset of $[0, 1]$ with positive measure. Show that there is $\alpha \in E$ and $\beta \neq 0$ such that $\alpha, \alpha + \beta, \alpha + 2\beta \in E$.

The only idea I have had is to use Lebesgue density and the Lebesgue Density Theorem, but so far, no luck.