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Consider sets $A\subseteq \mathbb{R}$ with Lebesgue measure zero and Hausdorff dimension one. For instance the set of real numbers with bounded entries in their continued fraction expansion have this property. Using iterated function systems it is easy to construct further examples.

I wonder I there is a quantity that distinguishes the "size" of such sets. The quantity should be a least monotone with respect to inclusion and stable with respect to unions.

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    $\begingroup$ But I doubt one can reach such a fine description through a single real number, maybe it would be necessary to define some kind of "scale-dependent" metric/curvature, hence generalizing Riemann ideas to much broader contexts. $\endgroup$ – Sylvain JULIEN Jul 9 '18 at 21:03
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    $\begingroup$ It would be interesting to see your example in more detail. For Brownian motion a similar issue happened and there we used a generalized Hausdorff dimension but with a gauge function that captures the dynamics. For many useful geometric measure theory results cf Mattila's geometry of sets and measures. For generalized H.dimension see chapter 4. $\endgroup$ – Thomas Kojar Jul 9 '18 at 21:24
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    $\begingroup$ One can of course refine the scale of (Hausdorff) measures by using scaling functions such as $h(t) = t(\log t)^{\alpha}$ etc. instead of $h(t)=t^{\alpha}$. $\endgroup$ – Christian Remling Jul 9 '18 at 21:25
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    $\begingroup$ And continue to refine ... $t \log(1/t)^\alpha\log\log(1/t)^\beta$ $\endgroup$ – Gerald Edgar Jul 9 '18 at 21:50
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    $\begingroup$ And more generally assigning a vector $ (\alpha_{i})_{i\geq 0} $ so that $ h(t)=\prod_{i}(\log_{i}(t))^{\alpha_{i}} $ with $ \alpha_{i} $ vanishing for all but finitely many $ i $ . $\endgroup$ – Sylvain JULIEN Jul 9 '18 at 21:54

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