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I am trying to find a set $A\subset\mathbb{R}$ with Hausdorff dimension $\log2/\log3=:s$ but has $H^s(A)=\infty$.

I suspect this is the Cantor set, but im struggling to show that it has Hausdorff measure (w.r.t $s$) $\infty$.

Any hints would be appreciated.

Bernard
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kam
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  • The Cantor set does not have infinite $\log_3(2)$-dimensional Hausdorff measure (I would say that the $\log_3(2)$-dimensional Hausdorff measure of the Cantor set is $1$, but this could be off by a constant, depending on the precise definition of the Hausdorff measure used), though some variation of the Cantor set is likely a good place to start looking for examples. – Xander Henderson May 28 '20 at 16:27
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    It might be helpful to think of more elementary examples first. Can you think of a set of Hausdorff dimension $1$ with infinite $1$-dimensional Hausdorff measure? What is property or properties might such a set have? Can you use that example to produce other examples with dimension other than $1$? – Xander Henderson May 28 '20 at 16:28
  • I think this is where my confusion lies. How can a set have Hausdorff dimension 1, but have 1-dimensional Hausdorff measure infinity? Isn't the Hausdorff dimension the infimum of $s>0$ s.t. the s-dimensional Hausdorff measure is 0? – kam May 28 '20 at 16:31
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    What is the Hausdorff dimension of the real line? What is its $1$-dimensional Hausdorff measure? – Xander Henderson May 28 '20 at 18:03

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