A density version of Cobham's theorem

TL;DR

This paper extends Cobham's theorem by showing that if two automatic sequences with independent bases agree on a set of density one, then they agree on a set with a periodic sequence, with applications to factorial digit problems.

Contribution

It introduces a density version of Cobham's theorem, strengthening the conditions under which automatic sequences must be ultimately periodic.

Findings

Proves a density-based extension of Cobham's theorem.

Shows automatic sequences agreeing on a density-one set are ultimately periodic.

Applies the theorem to factorial digit problems in number theory.

Abstract

Cobham's theorem asserts that if a sequence is automatic with respect to two multiplicatively independent bases, then it is ultimately periodic. We prove a stronger density version of the result: if two sequences which are automatic with respect to two multiplicatively independent bases coincide on a set of density one, then they also coincide on a set of density one with a periodic sequence. We apply the result to a problem of Deshouillers and Ruzsa concerning the least nonzero digit of $n!$ in base $12$.