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On asymptotically automatic sequences

Jakub Konieczny

TL;DR

This work introduces and develops asymptotically automatic sequences, extending the classical theory of automatic sequences by replacing exact kernel finiteness with finiteness up to almost-everywhere equality. It establishes closure properties, base-dependence phenomena, and a linear bound for the new notion of asymptotic subword complexity, while showing that asymptotic frequencies of symbols can fail to exist or be non-rational, unlike the automatic case. The authors provide classification results for bracket sequences and multiplicative sequences in the asymptotic setting and define asymptotically regular sequences, obtaining a Cobham-type theorem in that framework. Overall, the paper illuminates how density-zero perturbations interact with automata-inspired structure, yielding both parallels and notable departures from the classical theory with implications for density versions of automatic-structure results.

Abstract

We study the notion of an asymptotically automatic sequence, which generalises the notion of an automatic sequence. While $k$-automatic sequences are characterised by finiteness of $k$-kernels, the $k$-kernels of asymptotically $k$-automatic sequences are only required to be finite up to equality almost everywhere. We prove basic closure properties and a linear bound on asymptotic subword complexity, show that results concerning frequencies of symbols are no longer true for the asymptotic analogue, and discuss some classification problems.

On asymptotically automatic sequences

TL;DR

This work introduces and develops asymptotically automatic sequences, extending the classical theory of automatic sequences by replacing exact kernel finiteness with finiteness up to almost-everywhere equality. It establishes closure properties, base-dependence phenomena, and a linear bound for the new notion of asymptotic subword complexity, while showing that asymptotic frequencies of symbols can fail to exist or be non-rational, unlike the automatic case. The authors provide classification results for bracket sequences and multiplicative sequences in the asymptotic setting and define asymptotically regular sequences, obtaining a Cobham-type theorem in that framework. Overall, the paper illuminates how density-zero perturbations interact with automata-inspired structure, yielding both parallels and notable departures from the classical theory with implications for density versions of automatic-structure results.

Abstract

We study the notion of an asymptotically automatic sequence, which generalises the notion of an automatic sequence. While -automatic sequences are characterised by finiteness of -kernels, the -kernels of asymptotically -automatic sequences are only required to be finite up to equality almost everywhere. We prove basic closure properties and a linear bound on asymptotic subword complexity, show that results concerning frequencies of symbols are no longer true for the asymptotic analogue, and discuss some classification problems.
Paper Structure (13 sections, 33 theorems, 127 equations)

This paper contains 13 sections, 33 theorems, 127 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Basic properties
  4. Bases
  5. Frequencies of symbols
  6. Logarithmic frequencies
  7. Rationality of frequencies
  8. Subword complexity
  9. Classification problems
  10. Bracket sequences
  11. Multiplicative sequences
  12. Regular sequences
  13. Shift invariant multiplicative sequences

Key Result

Lemma 2.1

Let $k \geq 2$ and let $\mathbf a,\mathbf b$ be asymptotically $k$-automatic sequences. Then the Cartesian product $\mathbf a \times \mathbf b = \left((a_n,b_n)\right)_{n=0}^\infty$ is an asymptotically $k$-automatic sequence.

Theorems & Definitions (63)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 53 more