# Exact value of Hausdorff measure of middle-third Cantor set

Is there any result about the exact value of $\log_3 2$-dimensional Hausdorff measure of the middle-third Cantor set?

And is there any fractal (in $\mathbb R^n$) which is not contained in a $p$-dimensional space(hyperplane) and the exact value of the $\alpha$-dimensional (nonzero and finite) Hausdorff measure of the fractal with $0<\alpha < p$? (That is, the $p$-dimensional Hausdorff measure of the fractal is 0.)

## 1 Answer

The $\log_32$-measure of the standard Cantor set is $1$: this is proved in Theorem 1.14 of Falconer's book The Geometry of Fractal Sets.

More generally, whenever a self-similar Cantor-type set constructed from the unit interval has dimension $s$, its $s$-dimensional measure is also $1$. (Informally, it's inherited from the $s$-dimensional Hausdorff content of $[0,1]$, which is $1$.) This is Theorem 1.15 in the same book. Not reproducing the proofs as they are pretty long, and the book should be read anyway if one is interested in the subject.

To obtain examples not contained in a line, one can map the Cantor set to a semicircle by $t\mapsto \exp(i \pi t)$. This does not change the measure, because the diameters of short arcs are closely comparable to their arc length.

Alternatively, take the Cartesian square of a Cantor set. It is obtained from a unit square by repeatedly replacing it with smaller squares. The dimension doubles to $2s$, and the measure will be the $(2s)$-dimensional content of the unit square, which is $\sqrt{2}^{2s}$.