Let $X$ be a compact metric space, and let $$ \nu_s(X):=\sup\limits_{\varepsilon>0} \inf\limits_{\mathcal{E}} \sum\limits_{E \in \mathcal E} \mathrm{diam}(E)^s $$ be the $s$-dimensional Hausdorff measure of $X$, where $\mathcal E$ above ranges over covers of diameters less than $\varepsilon$. Then the Hausdorff dimension is defined as $$ \dim_H(X):=\inf\{ s: \nu_s(X)=0\}. $$

Some well-known facts:

  • Since $X$ is compact, we can take the infinum over finite open covers.
  • If we take the infinum over covers of the sets of the same diameter, we get Minkowski dimension instead.

My question is - is there any finer way to control the ratio of the diameters of balls in the finite open cover $\mathcal E$? Basically, given $s>0$ such that $\nu_s(X)=0$, I want to obtain a sequence of finite open covers $(\mathcal E_n)$ with diameters tending to zero such that $$ \sum\limits_{E \in \mathcal E_n} \mathrm{diam}(E)^s \to 0 $$ where the ratio $$ \frac{\text{diameter of the largest ball in } \mathcal E_n}{\text{diameter of the smallest ball in } \mathcal E_n} $$ does not grow too fast.

Of course, if we leave out the assumtion of compactness, the example of $\mathbb Q$ already tells us that, in general, this is not possible.

  • $\begingroup$ For the record, at time of writing this question has attracted a vote to close as "unclear what you're asking". I strongly disagree with this reason given $\endgroup$ – Yemon Choi May 8 '17 at 11:20

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