## A density version of Cobham’s theorem

### Volume 192 / 2020

Acta Arithmetica 192 (2020), 235-247
MSC: Primary 11B85; Secondary 11A63, 37B10, 68Q45, 68R15.
DOI: 10.4064/aa180626-13-1
Published online: 8 November 2019

#### 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 $12$.