From some Pisot numerations to topological groups

2026-06-29

AI Summary

This study addresses Pisot numeration systems satisfying the zero-inclusion property (Condition F) and extends them for the first time to the setting of p-adic integer groups, constructing an associated topological group ℤ_U. By integrating tools from algebraic number theory, linear recurrence sequences, and β-expansions, the authors establish a continuous group homomorphism from ℤ_U onto the torus. In particular, when the underlying numeration system is unimodular, they prove that ℤ_U is continuously isomorphic to the torus as a topological group. This work not only broadens the applicability of p-adic structures to non-integer bases but also uncovers a profound connection between digit representations in Pisot numeration systems and toral dynamics.

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.

Problem

• Pisot numeration
• topological groups
• zero-preserving
• Condition F
• p-adic integers

Innovation

• Pisot numeration
• topological group
• zero-preserving
• torus
• unimodular