An introduction to (alternate) numeration systems

In this talk, we introduce numeration systems as an object of study. After motivating the object, we introduce positional numeration systems (also called U-systems) and Rényi numeration systems (β-systems), two classical examples. We detail a correspondence between those two families of systems and recall how it can be used to study the regularity of the language of a U-system. We then explain how this correspondence can sometimes fail, and why the recently introduced alternate base numeration systems offer a solution, by replacing Rényi systems in the critical case. Finally, we present one specific result on these new numeration systems: with a set of "reasonable" expansions of 1, is associated exactly one alternate base.