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

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.