Hausdorff dimension of Cantor set times interval?

Question

What is the Hausdorff dimension of the set $C \times I \subset \mathbb{R}^2$ where $C$ is the standard Cantor set and $I$ is the unit interval? I know $dim_H (C) = \frac{ln2}{ln3}$ and I feel like there should be some easy argument here, but I'm not seeing one.

Answer 1

The usual argument for the cantor set uses self symmetry. The (middle thirds) Cantor set is constructed by cutting the unit interval into thirds, removing the middle section, repeating the construction to the remaining intervals and continuing the process. The set of points which are not removed in this process are precisely the points of $C$

In the first step of the process above (removing the middle third of the unit interval) you have two intervals remaining. These intervals naturally partition $C$ into a disjoint union $C_0\cup C_1=C$, where $C_0$ is the set of points remaining after continuing the process on the interval $[0,1/3]$ and $C^1$ is the same for $[2/3,1]$. If you consider the (Hausdorff) volume of these sets, notice that $V(C_0)=V(C_1)$, since $C_0$ is a copy of $C_1$. By the partition we described above, we therefore ought to have $$V(C)=V(C_0)+V(C_1)=2V(C_0)\;.$$ If you scale $C_0$ by a factor of 3, then by self similarity, you see you get $C$ again. In a $d$-dimensional space, volume scales as $\lambda^d$, where $\lambda$ is the scaling factor. For the Cantor set, that gives you $$V(C)=3^dV(C_0)\;.$$ Combining that with the above equation gives $$2V(C_0)=3^dV(C_0)\Rightarrow 2=3^d\;.$$ You solve that for the dimension and you get $d=\log(2)/\log(3)$.

Now, let's do the same trick for $I\times C$. Everything works the same, but when you scale by a factor of 3, you no longer get $C\times I$, you get $(C\times 3I)$. So then, $$3*2V(C_0\times I)=3^dV(C_0\times I)\Rightarrow 1+\log(2)/\log(3)=d\;,$$ which is what you expect.

Answer 2

Actually this is a special case of the following statement.

let $E \subset \mathbb{R}^n$. We have $$\dim(E \times I) = \dim(E) + 1$$

See either this post, or Theorem 5.1 and Theorem 5.8 in Falconer's The Geometry of Fractal Sets.