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I am going through the text 'Fractal Geometry: Mathematical Foundations and Applications' and came the following exercise:

Give a set $B\subset\mathbb{R}$ that has Hausdorff dimension $s=\frac{\log2}{\log3}$ but has $H^s(B)=\infty$.

My thoughts:

Now, when I see $\frac{\log2}{\log3}$ the first thing that comes to mind is the cantor set $C$, but the $H^s(C)$ is finite. I feel as though I am missing a lemma or theorem. Is there some relation of the Hausdorff dimension of a set constructed by countable unions?

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    $\begingroup$ If the union is nice enough (e.g. not two of the sets are nearer than a given $\delta > 0$), the Hausdorff dimension of the union is the supremum of the dimensions of the parts. Also note that Hausdorff measures are translation-invariant. $\endgroup$ – Daniel Fischer May 30 at 13:33
  • $\begingroup$ So ive I took a countable number of cantor sets, some $\delta>0$ apart, then the Hausdorff dimension would still be $s$ but, the Hausdorff measure $\infty$? $\endgroup$ – kam May 30 at 13:35
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    $\begingroup$ If your starting set has $H^s(C) > 0$. In this situation, the positive distance isn't needed (that's just a condition that works in every metric space), it suffices that there's no overlap. $\endgroup$ – Daniel Fischer May 30 at 13:43
  • $\begingroup$ math.stackexchange.com/questions/3686906/… looks similar $\endgroup$ – Claude May 30 at 15:10
  • $\begingroup$ math.stackexchange.com/questions/3695680 $\endgroup$ – Xander Henderson Jun 1 at 12:45

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