#### 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.

Done