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

2025-07-07

AI Summary

This paper addresses the problem of computing unified expansions of real numbers over infinitely many Cantor real bases, overcoming the limitation of conventional positional numeral systems that rely on fixed bases. The method introduces the first finite-state transducer capable of adaptively handling infinitely many varying bases, encoding the mapping from a real number and its associated base to the corresponding expansion as a single unified computational process. Theoretically, by integrating β-expansions, automata theory, and number-theoretic properties of Pisot numbers, we prove that when the base is a Pisot number satisfying certain algebraic conditions, the transducer operates with only finitely many states—ensuring computability and termination. Consequently, key combinatorial properties of expansions—such as greediness and eventual periodicity—become algorithmically decidable. This work establishes, for the first time, a computability framework for real-number expansions under variable bases.

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Abstract

Representing real numbers using convenient numeration systems (integer bases, $β$-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.

Problem

• Computing expansions in infinite Cantor real bases via transducer
• Deciding greediness and periodicity of expansions algorithmically
• Analyzing transducer structure for Pisot number bases

Innovation

• Single transducer computes expansions for infinite bases
• Finite transducer part visited under Pisot conditions
• Decidable greediness and periodicity in expansions