## A generalization of Cobham’s theorem. (Une généralisation du théorème de Cobham.)(French)Zbl 0814.11015

The author generalizes the notion of $$q$$-automaticity by defining $$\theta$$-automaticity where $$\theta$$ is a Pisot number. He then shows a Cobham-like theorem: the Cobham theorem [A. Cobham, Math. Syst. Theory 3, 186-192 (1969; Zbl 0179.025)] asserts that if a sequence is both $$k$$- and $$l$$-automatic, where $$\log k/\log l \in \mathbb{R}\setminus \mathbb{Q}$$, then this sequence is ultimately periodic.
In the paper under review the author shows that if a sequence is both $$q$$-automatic and $$\theta$$-automatic (where $$q$$ is an integer $$> 1$$ and $$\theta$$ a unitary Pisot number satisfying an extra assumption), then the sequence is ultimately periodic. He also indicates that the proof can be extended by replacing “$$\theta$$ unitary Pisot” by “$$\theta$$ Pisot and the constant term $$\alpha$$ of its minimal polynomial prime to $$q$$”. To have a generalization of Cobham’s theorem one would like to replace “$$\alpha$$ prime to $$q$$” by “$$\alpha$$ such that $$\log \alpha/\log q \in \mathbb{R}\setminus \mathbb{Q}$$”.

### MSC:

 11B85 Automata sequences 68R15 Combinatorics on words 68Q45 Formal languages and automata

Zbl 0179.025
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