A multi-dimensional analogue of Cobham's theorem for fractals

Let $\mathcal {C}_k$ be the set of hyper-cubes of size $\frac {1}{k} \times \cdots \times \frac {1}{k}$ in $[0,1]^d$ with vertices having coordinates of the form $(a_1/k, a_2/k, \ldots , a_d/k)$. For $c \in \mathcal {C}_k$ we define $T_c$ as the linear expansion map from $c$ to $[0,1]^d$ in the obvious way. We extend the map $T_c$ to a map on all of $[0,1]^d$ by defining $T_c(\overline {\mathbf {x}}) = \emptyset$ for $\overline {\mathbf {x}} \not \in c$.

Let $X$ be a compact subset of $[0,1]^d$ and $k > 1$ be an integer. We define the $k$-kernel of $X$ as

$$\{T_c(X) \ \mid \ c \in \mathcal {C}_1 \cup \mathcal {C}_k \cup \mathcal {C}_{k^2} \cup \mathcal {C}_{k^3} \cup \cdots \}.$$

If this set is finite, then we say that $X$ has finite $k$-kernel or, equivalently, that $X$ is $k$-self-similar. Some examples of this are the standard Cantor set, the Sierpiński carpet and the Sierpiński triangle.

Recently Adamczewski and Bell showed an analogue of Cobham's theorem for one-dimensional fractals. Let $k$ and $\ell$ be multiplicatively independent positive integers. They proved that the compact set $X \subset [0,1]$ is both $k$- and $\ell$-self-similar if and only if $X$ is a union of a finite number of intervals with rational endpoints.

In their paper, Adamczewski and Bell conjectured that a similar result should be true in higher dimensions. We prove their conjecture; in particular we prove:

Theorem. Let $k$ and $\ell$ be multiplicatively independent positive integers. The compact set $X \subset [0,1]^d$ is both $k$- and $\ell$-self-similar if and only if $X$ is a union of a finite number of rational polyhedra.