An Analogue of Cobham’s Theorem for Fractals

— We introduce the notion of k-self-similarity for compact subsets of R and show that it is a natural analogue of the notion of k-automatic subsets of integers. We show that various wellknown 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 R that are self-similar with respect to two multiplicatively independent bases k and l; namely, we show that X is both a kand a l-self-similar compact subset of R if and only if it is a finite union of closed intervals with rational endpoints.