Core Concepts

Sequences that are finite-state functions of the Zeckendorf numeration system can be characterized as solutions of generalized Mahler equations.

Abstract

The paper introduces a generalization of Mahler equations, called Z-Mahler equations, which characterize sequences that are finite-state functions of the Zeckendorf numeration system.
Key highlights:
The authors define Z-regular sequences as sequences that can be generated by weighted automata reading the Zeckendorf expansions of integers.
They introduce the concept of Z-Mahler equations, which are a generalization of classical Mahler equations to the Zeckendorf numeration system.
The main result is that any solution of an isolating Z-Mahler equation defines a Z-regular sequence, and conversely, any Z-regular sequence is the solution of a Z-Mahler equation.
The authors provide a construction of a weighted automaton that generates the solution of an isolating Z-Mahler equation.
They also show that the restriction to isolating equations is necessary, by providing an example of a non-isolating Z-Mahler equation whose solutions are not Z-regular.
The results generalize from the base-q numeration system to more general numeration systems generated by linear recurrences.

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Quotes

"We deﬁne generalised equations of Z-Mahler type, based on the Zeckendorf numeration system."
"If the Z-Mahler equation is isolating, then its solutions deﬁne Z-regular sequences."
"While the Zeckendorf analogue of n 7→qn is not linear, we show, by using work of Frougny [Fro92], that its non-linearity can be calculated by a deterministic automaton."

Key Insights Distilled From

by Olivier Cart... at **arxiv.org** 05-06-2024

Deeper Inquiries

The characterization of Z-regular sequences as solutions of Z-Mahler equations has several potential applications in various fields. One application is in the study of automatic sequences and their properties. By understanding the relationship between Z-regular sequences and Z-Mahler equations, researchers can further explore the properties and behaviors of automatic sequences in the context of Zeckendorf numeration. This can lead to advancements in areas such as combinatorics, number theory, and theoretical computer science.
Another potential application is in the field of transcendental number theory. The study of solutions to Mahler equations, including Z-Mahler equations, can provide insights into the transcendental properties of numbers generated by these equations. Understanding the transcendental nature of solutions to Z-Mahler equations can contribute to the broader understanding of transcendental numbers and their properties.
Furthermore, the characterization of Z-regular sequences as solutions of Z-Mahler equations can also have applications in cryptography and coding theory. Automatic sequences play a crucial role in these fields, and by leveraging the results of Z-Mahler equations, researchers can potentially develop new cryptographic techniques or coding algorithms based on the properties of Z-regular sequences.

Yes, the results obtained from the characterization of Z-regular sequences as solutions of Z-Mahler equations can be extended to numeration systems generated by other types of recurrences beyond linear ones. The general framework established in the study of Z-Mahler equations can be adapted to numeration systems generated by various types of recurrences, such as polynomial recurrences or more complex recursive formulas.
By generalizing the concepts and techniques developed for Z-Mahler equations to other types of recurrences, researchers can explore the properties of sequences generated by these numeration systems and their relationships to the solutions of corresponding Mahler equations. This extension can lead to a deeper understanding of automatic sequences in diverse mathematical contexts and pave the way for new discoveries in the field of number theory.

There are connections between the transcendental properties of solutions of Z-Mahler equations and those of classical Mahler equations. Mahler equations have been extensively studied in transcendental number theory due to their connection to transcendental numbers. The solutions of Mahler equations often exhibit interesting transcendental properties, such as being algebraic independent or having specific growth rates.
Similarly, solutions of Z-Mahler equations, which are associated with the Zeckendorf numeration system, can also possess transcendental properties. The study of Z-Mahler equations can reveal insights into the transcendental nature of numbers generated in the Zeckendorf numeration system and their relationships to classical Mahler equations.
By investigating the transcendental properties of solutions to Z-Mahler equations, researchers can draw parallels to the properties of solutions to classical Mahler equations and further enrich the understanding of transcendental numbers and their characteristics. This comparative analysis can lead to new discoveries and advancements in transcendental number theory.

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