S-automatic sets

In this talk, I will discuss some new properties of S-automatic sets. First I will show that a multidimensional set is S-automatic for all abstract numeration systems S if and only if it is 1-automatic. This result is surprising in the following sense: the class of multidimensional 1-automatic sets is a strict subclass of that of semi-linear sets. Hence, this result is not a generalization of the well-known result in integer base numeration systems: a multidimensional set is b-automatic for all integer bases b ≥ 1 if and only if it is semi-linear. Second I will describe the possible behaviors of the nth -term of an S-automatic set, depending on the growth function (i.e., the number of words of length n) of the numeration language.