An introduction to numeration systems: Cobham-like theorems, first-order logic and regular sequences

In this talk, I will present an introduction to numeration systems (mostly for representing integers), and show how first-order logic can be used in order to solve problems in combinatorics on words. The base idea is to translate properties of numbers into combinatorial properties of their representations. As far as I can, I will try to determine which results depends on the numeration systems involved and which do not. On the one hand, Cobham’s theorem and its generalizations tell us that most properties of numbers strongly depend on the chosen numeration system. On the other hand, the use of very general numeration systems, such as abstract numeration systems, allows us to understand how far we can exploit techniques from logic and automata theory. Along the way, we will define the notions of recognizable and definable sets of integers, automatic and regular sequences, morphic sequences and abstract numeration systems. If time allows me to do so, I will also present results generalizing these considerations to real numbers.