A density version of Cobham's theorem

Jakub Byszewski; Jakub Konieczny

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arXiv · 1710.07261

A density version of Cobham's theorem

Jakub Byszewski·Jakub Konieczny

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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 $12$.

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Jakub Byszewski, Jakub Konieczny. 2017-10-19. A density version of Cobham's theorem. https://arxiv.org/abs/1710.07261

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