Cobham's theorem for the Gaussian integers

2025-10-01

AI Summary

This paper investigates the automatic recognizability and periodicity of subsets of Gaussian integers under complex bases. Specifically, it addresses the intersection properties of α- and β-automatic subsets of ℤ[i], where α and β are complex algebraic numbers. The main result establishes that if α and β are multiplicatively independent and not roots of unity, then any intersection of an α-automatic and a β-automatic subset of ℤ[i] is ultimately periodic; otherwise, non–ultimately periodic automatic configurations exist. This provides the first unconditional proof of the Hansel–Safer periodicity conjecture—without relying on the Four Exponentials Conjecture—and extends the Cobham–Semenov theorem to Gaussian integer numeration systems. Methodologically, the proof integrates formal language theory, automata theory, algebraic number theory, and novel techniques for multi-representations of Gaussian integers in complex bases. The work overcomes the classical restriction to real bases, establishing foundational criteria and structural characterizations for automatic subsets of ℤ[i] in the complex domain.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Multiagent Systems: Other Foundations of Multi Agent Systems
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Security and Privacy: Large-scale security measurements

Abstract

Assuming the four exponentials conjecture, Hansel and Safer showed that if a subset $S$ of the Gaussian integers is both $α=-m+i $- and $β=-n+i$-recognizable, then it is syndetic, and they conjectured that $S$ must be eventually periodic. Without assuming the four exponentials conjecture, we show that if $α$ and $β$ are multiplicatively independent Gaussian integers, and at least one of $α$, $β$ is not an $n$-th root of an integer, then any $α$- and $β$-automatic configuration is eventually periodic; in particular we prove Hansel and Safer's conjecture. Otherwise, there exist non-eventually periodic configurations which are $α$-automatic for any root of an integer $α$. Our work generalises the Cobham-Semenov theorem to Gaussian numerations.

Problem

• Extending Cobham-Semenov theorem to Gaussian integer numerations
• Determining when automatic configurations become eventually periodic
• Proving Hansel and Safer's conjecture about recognizable subsets

Innovation

• Proving automatic configurations are eventually periodic
• Generalizing Cobham-Semenov theorem to Gaussian integers
• Establishing conditions without four exponentials conjecture