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

TL;DR

This paper presents a novel graph-theoretic proof of Cobham's Theorem, characterizing the growth of automatic sequences' support and providing new insights into their structure and rank.

Contribution

It introduces a new graph-theoretic approach to prove Cobham's Theorem and offers a fresh interpretation of the rank of sparse sequences using cycle arborescence height.

Findings

Support of automatic sequences is either sparse or grows at least like N^α.

The rank of sparse sequences correlates with the height of their cycle arborescence.

The maximum growth rate α is related to the logarithm of an integer root of a Perron number.

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.