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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 …

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 …

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.

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 …

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 …

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 $\dim_\mathrm{P}(B) = 2/5$, $ …

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

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 …

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 …

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 …

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 …