Computing Expansions in Infinitely Many Cantor Real Bases via a Single Transducer

TL;DR

This paper introduces a novel method using a single transducer to compute expansions of real numbers across an infinite family of Cantor real bases, enabling analysis of their properties and decidability of key features.

Contribution

It develops a unified transducer-based approach to handle infinitely many Cantor bases simultaneously, contrasting with traditional fixed-base models.

Findings

Finite transducer visits under certain Pisot number conditions

Decidability of greediness and periodicity of expansions

Structural analysis of the transducer for Cantor bases

Abstract

Representing real numbers using convenient numeration systems (integer bases, $\beta$-numeration, Cantor bases, etc.) has been a longstanding mathematical challenge. This paper focuses on Cantor real bases and, specifically, on automatic Cantor real bases and the properties of expansions of real numbers in this setting. We develop a new approach where a single transducer associated with a fixed real number $r$, computes the $\mathbf{B}$-expansion of $r$ but for an infinite family of Cantor real bases $\mathbf{B}$ given as input. This point of view contrasts with traditional computational models for which the numeration system is fixed. Under some assumptions on the finitely many Pisot numbers occurring in the Cantor real base, we show that only a finite part of the transducer is visited. We obtain fundamental results on the structure of this transducer and on decidability problems about these expansions, proving that for certain classes of Cantor real bases, key combinatorial properties such as greediness of the expansion or periodicity can be decided algorithmically.