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Asymptotically automatic sequences generalize the notion of automatic sequences by relaxing the requirement of finite k-kernels to finite up to asymptotic equality. This paper systematically investigates the properties of asymptotically automatic sequences and establishes analogues or counterparts to various well-known facts about automatic sequences.

Abstract

The paper studies the notion of asymptotically automatic sequences, which generalizes the concept of automatic sequences. While k-automatic sequences are characterized by the finiteness of their k-kernels, asymptotically k-automatic sequences only require the k-kernels to be finite up to asymptotic equality.
The key highlights and insights are:
Basic closure properties: Asymptotically automatic sequences are closed under Cartesian products, codings, and passing to arithmetic progressions.
Dependence on the base: For multiplicatively dependent bases, asymptotically automatic sequences are the same. However, for multiplicatively independent bases, there exist sequences that are asymptotically automatic in both bases without being asymptotically equal to a periodic sequence.
Frequencies of symbols: In contrast to automatic sequences, logarithmic frequencies of symbols in asymptotically automatic sequences are not guaranteed to exist, and if they do exist, they are not necessarily rational.
Subword complexity: Subword complexity is not the right notion to consider in the asymptotic regime. Instead, the author introduces the concept of asymptotic subword complexity, which is shown to be linear for asymptotically automatic sequences.
Classification problems: The paper fully classifies bracket words that are asymptotically automatic, showing that Sturmian words are never asymptotically automatic. It also partially classifies multiplicative sequences that are asymptotically automatic.
Asymptotically k-regular sequences: The author introduces the notion of asymptotically k-regular sequences and proves a variant of Cobham's theorem.

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Key Insights Distilled From

by Jakub Koniec... at **arxiv.org** 04-12-2024

Deeper Inquiries

The set of bases with respect to which a given sequence is asymptotically automatic can be further characterized by considering the structure of the set of bases itself. One approach is to investigate the relationships between the bases that yield asymptotically automatic sequences. For example, exploring the properties of multiplicatively dependent and independent bases in relation to asymptotic automaticity can provide insights into the classification of sequences based on their bases. Additionally, studying the distribution of frequencies of symbols across different bases can offer a deeper understanding of how the choice of base influences the asymptotic behavior of sequences. By analyzing the patterns and properties of bases that lead to asymptotically automatic sequences, researchers can classify sequences based on the characteristics of their associated bases.

Non-trivial examples of asymptotically automatic sequences that are not asymptotically equal to a periodic sequence or a sequence of the form an = F({logk n}) can be constructed by carefully designing the sequence to exhibit specific properties. One approach is to create sequences that have unique patterns or structures that do not align with periodicity or the logarithmic function. By manipulating the construction of the sequence and introducing complexities or irregularities in its behavior, it is possible to generate asymptotically automatic sequences that defy simple classification based on periodicity or specific functional forms. These examples showcase the diversity and richness of asymptotically automatic sequences beyond conventional patterns, highlighting the complexity and versatility of this class of sequences.

Asymptotically automatic sequences have deep connections to various areas of mathematics, including dynamical systems, ergodic theory, and number theory. In dynamical systems, these sequences can be interpreted as symbolic sequences generated by dynamical processes, providing insights into the long-term behavior and properties of the underlying systems. Ergodic theory offers a framework to study the statistical properties of these sequences, such as their entropy and mixing properties, shedding light on their complexity and randomness. In number theory, asymptotically automatic sequences can be linked to Diophantine approximation, transcendence theory, and other number-theoretic concepts, offering new perspectives on the interplay between arithmetic properties and automaticity. By exploring these connections, researchers can uncover deeper relationships between asymptotically automatic sequences and fundamental mathematical theories, enriching the understanding of these sequences and their implications across different mathematical disciplines.

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