shift power mod 1 of the cantor set by an irrational number and their intersections

Question

Let $C$ be the Cantor ternary set and consider the shift $T_a$ mod 1 of the interval $[0,1]$ for an irrational number $a\in[0,1]$. I'm wondering whether $T_a^k(C)\cap T_a^l(C)=\emptyset$, $k,l\in {\Bbb Z}$, $k\neq l$.

Answer 1

Observe that for $k\in\Bbb Z$ we have $T_a^k=T_{ka}$. Thus, the question boils down to asking whether $T_{ka}[C]\cap T_{\ell a}[C]=\varnothing$, which is equivalent to asking whether $T_{(k-\ell)a}[C]\cap C=\varnothing$. We might as well simply ask whether $T_a[C]\cap C=\varnothing$. Let $\alpha=a-\lfloor a\rfloor$, the fractional part of $a$. It’s well-known that $C-C=[-1,1]$, so $\alpha\in C-C$, i.e., there are $x,y\in C$ such that $\alpha=x-y$. But then

$$x=y+\alpha\equiv(y+a)\!\!\!\pmod1\;,$$

so $x\in T_a[C]\cap C\ne\varnothing$.