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Math Help - Hausdorff dimension of sets

  1. #1
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    Hausdorff dimension of sets

    What is the Hausdorff dimension of the following sets?

    a) { (x,x) R^2 : x Q } where R is the real numbers and Q is the rational numbers.

    b) { (x,x) R^2 : x R }

    c) { (x,x)R^2 : x C } where C is the middle third cantor set.



    I think the answer to c) is log2/log3 but i am not sure how to find the other two Hausdorff dimensions.

    Thanks.
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  2. #2
    MHF Contributor

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    Perhaps because they are so trivial!
    The Hausdorf dimension of the points in R^2, both of whose coordinates are rational, is 0 because that set is countable.

    The Hausdorf dimension of all of the points in R^2 is 2.
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