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.
$\begingroup$
$\endgroup$
2
-
2$\begingroup$ This addresses your question, essentially: cut-the-knot.org/do_you_know/cantor3.shtml The difference is that $x=y$ is allowed. In this case, the first difference set is already the whole interval $[-1,1]$. $\endgroup$– Alessandro Della CorteCommented Sep 17, 2022 at 15:03
-
$\begingroup$ Thank you @Alessandro Della Corte for the link.Btw,it is distressingly difficult to come up with an interesting enough problem that somebody has not already posed.When it came to my mind i felt a little euphoria that i posed a little interesting problem knowing not it is already there. $\endgroup$– AgnostMysticCommented Sep 18, 2022 at 6:26
Add a comment
|