Linear recursive odometers and beta-expansions

TL;DR

This paper explores the relationship between properties of beta-expansions and odometers, focusing on conditions that ensure pure discrete spectrum, with implications for Pisot numerations and tilings.

Contribution

It establishes the connection between Hypothesis B and the (QM) condition, advancing understanding of spectral properties in beta-expansions and odometers.

Findings

Hypothesis B implies (QM) for G-odometers.

Both conditions ensure pure discrete spectrum.

Results apply to Pisot beta-numerations and related tilings.

Abstract

The aim of this paper is to study the connection between different properties related to $\beta$-expansions. In particular, the relation between two conditions, both ensuring pure discrete spectrum of the odometer, is analysed. The first one is the so-called Hypothesis B for the $G$-odometers and the second one is denoted by (QM) and it has been introduced in the framework of tilings associated to Pisot $\beta$-numerations.