An analogue of Cobham’s theorem for fractals

Abstract:

We introduce the notion of $k$-self-similarity for compact subsets of $\mathbb {R}^n$ and show that it is a natural analogue of the notion of $k$-automatic subsets of integers. We show that various well-known fractals such as the triadic Cantor set, the Sierpiński carpet or the Menger sponge turn out to be $k$-self-similar for some integers $k$. We then prove an analogue of Cobham’s theorem for compact sets of $\mathbb R$ that are self-similar with respect to two multiplicatively independent bases $k$ and $\ell$. Namely, we show that $X$ is both a $k$- and an $\ell$-self-similar compact subset of $\mathbb {R}$ if and only if it is a finite union of closed intervals with rational endpoints.