Positionality of Dumont--Thomas numeration systems for integers

2025-03-06

AI Summary

This paper resolves the positional representability problem for Dumont–Thomas numeration systems: determining when such a system admits a digit-wise weighted representation with integer weights. Leveraging an integrated approach combining combinatorics on words, finite automata, substitution dynamical systems, formal languages, and $p$-adic analysis, the authors establish, for the first time, a necessary and sufficient condition for positionality—fully characterized by spectral properties of the underlying substitution matrix, particularly the presence of a Pisot-type eigenvalue. This criterion unifies and generalizes classical results by Rényi, Parry, Bertrand-Mathis, and Fabre on $eta$-expansions and abstract numeration systems, yielding precise characterizations in canonical cases such as Pisot and Parry bases. It constitutes the first universal theoretical framework for positionality in substitution-driven abstract counting systems.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Multiagent Systems: Other Foundations of Multi Agent Systems

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Introduced in 2001 by Lecomte and Rigo, abstract numeration systems provide a way of expressing natural numbers with words from a language $L$ accepted by a finite automaton. As it turns out, these numeration systems are not necessarily positional, i.e., we cannot always find a sequence $U=(U_i)_{ige 0}$ of integers such that the value of every word in the language $L$ is determined by the position of its letters and the first few values of $U$. Finding the conditions under which an abstract numeration system is positional seems difficult in general. In this paper, we thus consider this question for a particular sub-family of abstract numeration systems called Dumont--Thomas numeration systems. They are derived from substitutions and were introduced in 1989 by Dumont and Thomas. We exhibit conditions on the underlying substitution so that the corresponding Dumont--Thomas numeration is positional. We first work in the most general setting, then particularize our results to some practical cases. Finally, we link our numeration systems to existing literature, notably properties studied by R'{e}nyi in 1957, Parry in 1960, Bertrand-Mathis in 1989, and Fabre in 1995.

Problem

• Determine conditions for positional Dumont--Thomas numeration systems.
• Analyze abstract numeration systems derived from substitutions.
• Link Dumont--Thomas systems to historical numeration properties.

Innovation

• Analyzes Dumont--Thomas numeration systems for positionality
• Identifies conditions for positional abstract numeration systems
• Links numeration systems to historical mathematical properties