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arXiv · 2606.30496

From some Pisot numerations to topological groups

Olivier Carton·Jake Sudbery·Reem Yassawi
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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 $\beta$-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.

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Olivier Carton, Jake Sudbery, Reem Yassawi. 2026-06-29. From some Pisot numerations to topological groups. https://arxiv.org/abs/2606.30496

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