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The Sum of 2 sets of Measure zero might well be very large for example, the sum of $x$-axis and $y$-axis, is nothing but the whole plane. Similarly one can ask this question about Cantor sets:

If $C$ is the cantor set, then what is the measure of $C+C$?

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Here is the full question from Halmos's Problems for mathematicians, young and old: alt text

Here's the hint:

alt text

And here's the solution:

alt text

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Were you ever a detective in a previous life? No $h*t. :) – user1119 Dec 11 '10 at 10:38
@George S.: In this case, I just happen to be a Halmos fan who is also a fan of citing sources. – Jonas Meyer Dec 11 '10 at 19:17
up vote 19 down vote accepted

If you are asking the case where $C$ is the Cantor ternary set, then you can show that $C+ C$ is actually $[0,2]$.

For more general cantor sets, you can find a description in the paper: On the topological structure of the arithmetic sum of two Cantor sets, P Mendes and F Oliveira, available at .

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George S: Excellent article man! Thanks +1 – anonymous Aug 14 '10 at 9:16
@Chandru: No problem. You are welcome. :) – user1119 Aug 14 '10 at 9:20

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