Let $C$ be the Cantor ternary set obtained by repeatedly deleting the middle thirds of the interval $[0,1].$ Now define $$E_1=C$$ and $$E_{i+1}=\{ |x-y|,x \neq y\text{ and } \,x,y \in E_i \}$$ I wonder what we can say about the properties of $E$ where $$ E=\lim_{n \to \infty} E_n .$$ Is $E$ a set of zero or nonzero measure? Can it include some rational number? I have no idea but the question seems interesting to me.
I would be highly obliged for any insights/observations/hints/answers.