Alladi Ramakrishnan Hall

Cobham’s theorem

J-M. Deshouillers

University of Bordeaux

These lectures present a proof of Cobham’s theorem stated below. They represent the
last part of the lectures I gave at IMSc on Automatic sequences. However, this set
of lectures is self-contained: no \emph{a priori} knowledge of automatic sequences
is required.
Cobham’s theorem states that if an infinite set $X$ of integers is both $k$ and
$\ell$ automatic, where $\log k$ and $\log \ell$ are $\mathbb{Q}$-linearly
independent, then $X$ is ultimately periodic.
In lecture 1, we recall definitions and basic facts about automatic sequences and we
present a result on Diophantine approximation.
In lecture 2, we show that a sequence satisfying the hypothesis of Cobham’s theorem
is syndetic, i.e. the difference between its consecutive terms is bounded.
In lecture 3, we end the proof of Cobham’s theorem.