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Let $K$ be a compact set in $\mathbb{R}^d$ of Hausdorff dimension $\alpha<d$, $H_\alpha(\cdot)$ the $\alpha$-dimensional Hausdorff measure. If $0<H_\alpha(K)<\infty$, is it necessarily true that $H_\alpha(K\cap B)\lesssim r(B)^\alpha$ for any open ball $B$? Here $r(B)$ denotes the radius of the ball $B$.

This seems to be true when $K$ enjoys some self-similarity, e.g. when $K$ is the standard Cantor set. But I am not sure if it is also true for the general sets.

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In case you want to study this subject further, I'll point out the key words: (a) "Ahlfors regular" means $\mathcal H^{\alpha}(K\cap B_r)\sim r^{\alpha}$; (b) "upper Ahlfors regular" means $\mathcal H^{\alpha}(K\cap B_r)\lesssim r^{\alpha}$. Some people drop "Ahlfors". – user31373 May 25 '12 at 22:43
up vote 3 down vote accepted

Consider e.g. $\alpha=1$, $d=2$. Given $p > 1$, let $K$ be the union of a sequence of line segments of lengths $1/n^2$, $n = 1,2,3,\ldots$, all with one endpoint at $0$. Then for $0 < r < 1$, if $B$ is the ball of radius $r$ centred at $0$, $H_1(K \cap B) = \sum_{n \le r^{-1/2}} r + \sum_{n > r^{-1/2}} n^{-2} \approx r^{1/2}$

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Nice! Thank you. – Syang Chen May 26 '12 at 14:11

I would read about the local dimension of a measure here.

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