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Suppose we construct sets $A,B,C$ with $A \subseteq [0,1]$, $\dim_\mathrm{H}(A) = 1/5$ and $\dim_\mathrm{P}(A) = 4/5$, $B \subseteq [2,3]$, $\dim_\mathrm{H}(B) = 1/5$ and $\dim_\mathrm{P}(B) = 2/5$, $ …

A construction used (repeatedy) in the paper Edgar, G. A., Centered densities and fractal measures, New York J. Math. 13, 33-87 (2007). ZBL1112.28004. For more information, see that paper. We constru …

Not true for all $X$. Taking the gauge function $\phi(t) = t^{1/2}/\log|t|$, construct a Cantor set $X$ in $\mathbb R$ using Hausdorff's original method so that $H^\phi(X) = 1$. Then the Hausdorff d …

One example: the set of Liouville numbers has Hausdorff dimension zero. In number theory, a Liouville number is an irrational number $x$ with the property that, for every positive integer $n$, the …

The first reference is Hausdorff's paper, "Dimension und Äußeres Maß" He constructs a Cantor set $E \subseteq [0,1]$ with $0 < \mathcal H^h(E) < +\infty$ for any sensible $h$. So if $0 < s < 1$ this …

According to Falconer[1] this is due to Besicovitch[2]. Falconer states it (generalized to $\mathbb R^n$) as Theorem 5.1, p. 65. This proves more than just $X$ and $\pi(X)$ have the same Hausdorff d …

K. Falconer, Fractal Geometry: Mathematical Foundations and Applications (3rd ed, 2014, Wiley). page 77 Corollary 4.12 Let $F$ be a Borel subset of $\mathbb R^n$ with $0 < \mathcal H^s(F) \le \inf …

See my paper LINK Centered densities and fractal measures, New York Journal of Mathematics 13 (2007) 33-87 Some references are also at the end of it. In particular, Boardman, Goodey, and McClure.

My question is whether there are nowhere differentiable continuous functions (between real numbers) whose graph are not fractals? Of course this depends on your definition of fractal. There are no …

As Charles said, this would be the box-counting dimension (a.k.a. Bouligand dimension or Minkowski dimension). Not the Hausdorff dimension. And for the measure it would usually not converge so that …

The answer is 2. Besicovitch and Ursell, Sets of fractional dimensions (V): On dimensional numbers of some continuous curves. J. London Math. Soc. 12 (1937) 18–25. doi:10.1112/jlms/s1-12.45.18

As stated, countable sets have Hausdorff dimension 0. So any set $S$ with $\mathrm{HD}(S)>0$ has power $\ge \aleph_1$. No need for continuum hypothesis. Without CH, though, we cannot say whether powe …