Call a Borel set $A \subseteq [0,1]$ *good* if $$0 < \dim(A) \leq \overline{\dim\_\text{M}}(A) < 2 \dim(A),$$ where $\dim(A)$ is the Hausdorff dimension of $A$ and $\overline{\dim\_\text{M}}(A)$ is the upper Minkowski dimension of $A$. (Note that the second inequality in the string of three inequalities holds automatically.)



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> Does a compact set $X \subseteq [0,1]$ of positive Hausdorff dimension necessarily contain a subset which is a *good* set?
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A positive answer leads to some more questions: Is there a subset of $X$ with the same Hausdorff dimension as $X$ which is also *good*? To what extent can the $2$ in the definition of *good* be reduced while maintaining a positive answer? Is it possible to find a subset for which the Hausdorff and upper Minkowski dimensions agree?


Otherwise, I'm hoping for a counterexample: a set with positive Hausdorff dimension with the property that all positive-dimension subsets have "large" upper Minkowski dimension. Given such an example, would it further be possible to have an $\epsilon > 0$ such that all positive-dimension subsets have upper Minkowski dimension greater than $\epsilon$?



Subsets of sets of positive Hausdorff dimension with controlled upper Minkowski dimension
A counterexample:


Let $E = \{0,1\}$ and let $\pi:E^{\mathbb N}\to [0,1]$ be the coding map for the Cantor set. Now fix $N\in\mathbb N$, let
$$
A = \{n\in\mathbb N : \lfloor \log\_N(n)\rfloor \text{ is even}\},
$$
let $S = \{\omega\in E^{\mathbb N} : \omega\_n = 0 \; \forall n\in A\}$, and let $X = \pi(S)$. A standard computation (e.g. mass distribution principle on the Bernoulli measure on $X$) shows that
$$
\dim(X) = \log\_3(2)\sum\_{n\text{ odd}} \left[\frac{1}{N^n} - \frac{1}{N^{n + 1}}\right] = \frac{\log\_3(2)}{N + 1} > 0.
$$
Now let $A$ be a subset of $X$, and fix $s < \dim(A)$. By Frostman's lemma, there exists a probability measure $\mu$ supported on $A$ such that for all $B(x,r)$, $\mu(B(x,r)) \leq C r^s$, where $C > 0$ is fixed. Now fix an even integer $n$ and let
$$
r\_n = 3^{-N^n}, \;\; r\_{n + 1} = 3^{-N^{n + 1}}.
$$
Then for all $x\in X$, $B(x,r\_n)\cap X = B(x,r\_{n + 1})\cap X$ (ignoring endpoints), so
$$
\mu(B(x,r\_n)) \leq C r\_{n + 1}^s.
$$
So if $B\subseteq A$ is an $r\_n$-net, then
$$
\#(B) \geq \frac{1}{C r\_{n + 1}^s} = \frac{1}{C} r\_n^{-Ns}.
$$
Thus $\overline{\dim\_M}(A) \geq Ns$; since $s$ was arbitrary,
$$
\overline{\dim\_M}(A) \geq N \dim(A).
$$


