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arXiv · 2405.11385

A graph-theoretic proof of Cobham's Dichotomy for automatic sequences

Mieke Wessel
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Abstract

We give a new graph-theoretic proof of Cobham's Theorem which says that the support of an automatic sequence is either sparse or grows at least like $N^\alpha$ for some $\alpha > 0$. The proof uses the notions of tied vertices and cycle arboressences. With the ideas of the proof we can also give a new interpretation of the rank of a sparse sequence as the height of its cycle arboressence. In the non-sparse case we are able to determine the supremum of possible $\alpha$, which turns out to be the logarithm of an integer root of a Perron number.

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Mieke Wessel. 2024-05-18. A graph-theoretic proof of Cobham's Dichotomy for automatic sequences. https://arxiv.org/abs/2405.11385

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