Hausdorff dimension of the Cantor Set

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SUMMARY

The Hausdorff dimension of the Cantor set is definitively calculated as \(\frac{\log 2}{\log 3}\). To prove this, one must utilize the definition of Hausdorff dimension, which involves demonstrating that the dimension is less than or equal to \(\frac{\log 2}{\log 3}\) in one part and greater than or equal to \(\frac{\log 2}{\log 3}\) in another. This two-part proof is essential for establishing the exact value of the dimension.

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  • Understanding of Hausdorff dimension
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alpar_r
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Hi everyone!

I am thinking about, how can calculate the Hausdorff dimension of the Cantor set? I know, that this dimension is \frac{\log 2}{\log 3} but I cannot prove it.

Any ideas?
 
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Of course you will need to use the definition of Hausdorff dimension. In one popular proof, you do it in two parts: One shows the dimension is <= log 2/log 3 and the other shows the dimension is >= log 2/log 3 . But of course both of them need the definition!
 

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