This question arises from a previous answer of mine, in which I claimed a fact I am not so sure about.
Let $\alpha\in(1,2)$ be an irrational number (feel free to assume $\alpha=\sqrt{2}$ or $\alpha=\frac{1+\sqrt{5}}{2}$ if you are more comfortable with periodic continued fractions) and let $E$ be the set of numbers of the form $\lfloor n\alpha\rfloor$ with $n\in\mathbb{Z}$.
Claim: there exists $F\subset\mathbb{Z}$ with positive lower asymptotic density such that the set $F-F$ (Minkowski difference) is entirely contained in $E$.
I thought it was enough to exploit the fact that the discrete Fourier transform of $\mathbb{1}_E$ is well-behaved (so Szemerédi's theorem applies, for instance), but I am having troubles in actually achieving a positive density for $F$. Maybe I am regarding this the wrong way, and it is enough to remove few elements from $E$ and get $\overline{E}$ such that $\overline{E}-\overline{E}\subset E$, in a purely combinatorial fashion. Any help?