Bridges between Automatic Sequences, Algebra and Number Theory

School: 24 - 28 April 2017
Workshop: 1 - 5 May 2017

Organizers : School - Valérie Berthé (Institut de Recherche en Informatique Fondamentale), Sébastien Labbé (CNRS); Workshop - Boris Adamczewski (CNRS & Université de Lyon), Jason Bell (University of Waterloo)


The school will consist of mini-courses on topics issued from combinatorics of words, symbolic dynamics, and on some of the algebraic and number theoretic background that will be useful for participants of the Workshop Bridges between Automatic Sequences and Algebra and Number Theory.

The main invited speakers are

• Boris Adamczewski (CNRS & Université de Lyon) • Yann Bugeaud (IRMA, Strasbourg, France) • Christophe Reutenauer (UQAM, Montréal, Canada) • Reem Yassawi (IRIF, Paris, France)

The school will include a mini-course by Christophe Reutenauer on the correspondence between Markoff numbers and Christoffel words.

Boris Adamczewski and Reem Yassawi will give an introduction to automatic sequences including fundamental properties (definitions, frequencies, morphisms, kernel, automatic set of integers, importance of the base, ...) then show Christol's theorem, and Furstenberg's theorem on diagonals, Skolem-Mahler-Lech results and the link with Mahler equations.

Yann Bugeaud will give a minicourse on complexity, automatic numbers and Diophantine approximation.

Other invited speakers will give short talks:

• Robbert Fokkink (TU Delft, Netherlands) • Julien Leroy (Université de Liège, Belgium) • Narad Rampersad (University of Winnipeg, Canada) • Eric Rowland (Hofstra University, USA) • Št?pán Starosta (Czech Technical University in Prague, Czech Republic)

There will be organized computer exercises and experiments sessions every day allowing to develop intuitions on the presented mini-courses and talks.


The theory of finite-state automata and automatic sequences has recently seen numerous applications to algebra and number theory. This is evidenced by the work of Derksen and Masser addressing solutions to S-unit equations in positive characteristic, work on transcendence and irrationality exponents of Mahler functions and automatic real numbers done by many: Adamczewski, Bugeaud, Bell, Cassaigne, Coons, Zudilin, and others over the past ten years.

In addition, there has been some suggestion that methods of automata could be used to obtain an effective proof of the Mordell-Lang conjecture in positive characteristic. Currently, a non-effective proof exists using model theoretic methods of Hrushovski. Work of Moosa and Scanlon suggests that at least part of this proof could be made effective by studying Frobenius actions carefully and associating an appropriate automaton to the given geometric data.