Automatic semigroups

Abstract: Automatic sequences, that is, sequences computable by finite automata, have been extensively studied from a variety of perspectives, including combinatorics, number theory, dynamics and theoretical computer science. Classification problems are a natural class of questions in the theory of automatic sequences. In particular, the problem of classifying automatic multiplicative sequences has attracted considerable attention, culminating in complete classification which we obtained in joint work with Clemens M\"{u}llner and Mariusz Lema\'{n}czyk. The subject of my talk will be an extension of this line of inquiry, which we pursue in joint work with Oleksiy Klurman. Under mild technical assumptions, we classify all automatic multiplicative semigroups, that is, all sets $E$ of integers which are closed under multiplication and such that the indicator function $1_E$ is automatic. Additionally, we show (again, under mild technical assumptions) that if $E,F$ are automatic sets with $E \cdot F \subset E$ then $E$ must contain a large essentially periodic component. This leads to potentially interesting open problems concerning products of automatic sets.

number theory

Audience: researchers in the discipline

Combinatorial and additive number theory (CANT 2022)