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Is there any way to show that the set of disjoint translations of the cantor ternary set is countable?

That is show that there are countably many disjoint sets of the form $\{x+C: x\in \mathbb{R}\}$???

Thanks

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    Don't we have $x + C \cap y + C \ne \emptyset$ iff $x-y \in C- C = [-1,1]$ and hence only countably many disjoint translates?2012-09-20

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