From some Pisot numerations to topological groups
2026-06-29 • Formal Languages and Automata Theory
Formal Languages and Automata Theory
AI summaryⓘ
The authors study a special way of counting numbers called a Pisot numeration system, which is built using a certain type of recurrence related to Pisot numbers. They create a new mathematical object similar to p-adic integers but designed for these systems when they meet a specific 'preserving zeros' condition. Their work shows that these objects, called \( \mathbb{Z}_U \), can be mapped onto tori (donut-shaped spaces) in a nice, structure-preserving way. Moreover, if the system is unimodular, \( \mathbb{Z}_U \) behaves exactly like a torus in terms of its topology and group structure.
Pisot numeration systemPisot numberlinear recurrencep-adic integersCondition Ftopological grouptorusunimodularhomomorphismβ-numerations
Authors
Olivier Carton, Jake Sudbery, Reem Yassawi
Abstract
A Pisot numeration system $U$ for $\mathbb N$ is a sequence of natural numbers generated by an integral homogeneous linear recurrence whose characteristic polynomial is the minimal polynomial of a Pisot number. The purpose of this paper is to introduce the analogue of the group of $p$-adic integers for such numerations when they \emph{preserve zeros}, which is equivalent to the `Condition F' introduced by Frougny and Solomyak for $β$-numerations. We show that these topological groups $\mathbb Z_U$ project homomorphically onto a torus. Equipping $\mathbb Z_U$ with the appropriate topology, we also show that if $U$ is unimodular, then $\mathbb Z_U$ is continuously isomorphic to a torus.