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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 $T_a^k(C)\cap T_a^l(C)=\emptyset$, $k,l\in {\Bbb Z}$, $k\neq l$.

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  • $\begingroup$ To be sure that I understand your notation: isn’t $T_a^k$ just $T_{ka}$? $\endgroup$ – Brian M. Scott Nov 15 '14 at 17:39
  • $\begingroup$ $T^k_a=(T_a)^k$, the $k$-power $\endgroup$ – francesco fidaleo Nov 15 '14 at 17:41
  • $\begingroup$ That’s what I thought. Isn’t that the same as $T_{ka}$? $\endgroup$ – Brian M. Scott Nov 15 '14 at 17:41
  • $\begingroup$ I don't know, maybe not because of "mod 1", consider $C$ as living on the circle $\endgroup$ – francesco fidaleo Nov 15 '14 at 17:44
  • $\begingroup$ I was taking that into account. You’re shifting it $a$ units around the circle $k$ times. That’s the same as shifting it $ka$ units once. $\endgroup$ – Brian M. Scott Nov 15 '14 at 17:48
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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 $x\in T_a[C]\cap C\ne\varnothing$.

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  • $\begingroup$ everything is clear now (thank you), but I've the following related question: it is known the cardinality of $T_a(C)\cap C$ when $a\in[0,1]$ is irrational? $\endgroup$ – francesco fidaleo Nov 15 '14 at 18:55
  • $\begingroup$ @francesco: You’re welcome. That’s a good question; I don’t offhand know the answer, but I’d almost bet that it’s $2^\omega=\mathfrak{c}$. The intersection is a closed set, so it must be countable or have cardinality $2^\omega$, and I’d be very much surprised if it could be countable. $\endgroup$ – Brian M. Scott Nov 15 '14 at 18:59
  • $\begingroup$ yes Brian, thank you. Another question is (in case it is not countable): what is $\mu(T_a(C)\cap C)$ where $\mu$ is the Cantor ternary probability measure (still in case $a$ irrational) $\endgroup$ – francesco fidaleo Nov 15 '14 at 19:03
  • $\begingroup$ @francesco: I’d have to give that some serious thought, and if I have time, I will. I suggest, though, that you post it as a separate question, so that more people see it. $\endgroup$ – Brian M. Scott Nov 15 '14 at 19:28
  • $\begingroup$ Thank you Brian, keep in contact, if I find something about that I'll keep you informed $\endgroup$ – francesco fidaleo Nov 15 '14 at 22:41

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