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.
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 $0$, $H_1(K \cap B) = \sum_{n \le r^{-1/2}} r + \sum_{n > r^{-1/2}} n^{-2} \approx r^{1/2}$