Project III

Automatic sequences in number theory and beyond

Gabriel Fuhrmann

Description. Roughly speaking, automatic sequences are infinite sequences over some finite alphabet which are generated through some finite automaton (for details, see the below reference). Quite intriguingly, they appear all over the place in mathematics (number theory, ergodic theory, fractal geometry) or even theoretical physics (most notably, in the context of certain Schrödinger Operators).

In a way, the finite character (inherited from the finite automaton) of these infinite objects places them somewhere at the interface between order and disorder or chaos. Besides their intrinsic beauty, this is one of the reasons why they are particularly well studied.

There is a vast and seemingly infinite amount of literature available but a very good and accessible (still quite comprehensive) starting point is [1]. We will start the project by covering the basics together: shift spaces, finite automata, some basic number theory and then, automatic sequences themselves. Afterwards, everyone can branch off towards topics they are particularly interested in. In this way, nobody will get lost and everyone can still follow their own interests.

Corequisites. Nothing is really necessary as everything can be covered to the extent it is needed in this project. It may help if you’re taking Topology 3 or Analysis 3 but don’t worry if you aren't.

Literature.

[1]     Jean-Paul Allouche and Jeffrey Shallit. Automatic sequences:           Theory, Applications, Generalizations. Cambridge University           Press, Cambridge, 2003.