Cobham's Theorem and Automaticity

TL;DR

This paper makes bounds in Krebs' proof of Cobham's theorem explicit, providing upper bounds on common prefix lengths of certain automatic sequences and showing limitations on their overlap with Sturmian sequences.

Contribution

It explicitly bounds the common prefix length in Krebs' proof of Cobham's theorem and demonstrates that automatic sequences cannot share arbitrarily large factors with Sturmian sequences.

Findings

Explicit bounds on common prefix lengths for automatic sequences

Automatic sequences cannot have arbitrarily large factors in common with Sturmian sequences

Enhanced understanding of the structure of automatic and Sturmian sequence overlaps

Abstract

We make certain bounds in Krebs' proof of Cobham's theorem explicit and obtain corresponding upper bounds on the length of a common prefix of an aperiodic $a$-automatic sequence and an aperiodic $b$-automatic sequence, where $a$ and $b$ are multiplicatively independent. We also show that an automatic sequence cannot have arbitrarily large factors in common with a Sturmian sequence.