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Arithmetical subword complexity of automatic sequences

Jakub Konieczny, Clemens Müllner

TL;DR

These are the automatic sequences with the maximal possible arithmetical subword complexity of a given automatic sequence a with the property that each word over $\Omega$ appears is $a$ along an arithmetic progression.

Abstract

We fully classify automatic sequences $a$ over a finite alphabet $Ω$ with the property that each word over $Ω$ appears is $a$ along an arithmetic progression. Using the terminology introduced by Avgustinovich, Fon-Der-Flaass and Frid, these are the automatic sequences with the maximal possible arithmetical subword complexity. More generally, we obtain an asymptotic formula for arithmetical (and even polynomial) subword complexity of a given automatic sequence $a$.

Arithmetical subword complexity of automatic sequences

TL;DR

These are the automatic sequences with the maximal possible arithmetical subword complexity of a given automatic sequence a with the property that each word over appears is along an arithmetic progression.

Abstract

We fully classify automatic sequences over a finite alphabet with the property that each word over appears is along an arithmetic progression. Using the terminology introduced by Avgustinovich, Fon-Der-Flaass and Frid, these are the automatic sequences with the maximal possible arithmetical subword complexity. More generally, we obtain an asymptotic formula for arithmetical (and even polynomial) subword complexity of a given automatic sequence .
Paper Structure (8 sections, 13 theorems, 52 equations)

This paper contains 8 sections, 13 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Automata
  4. Higher order Fourier analysis
  5. Lower bound
  6. Upper bound
  7. Primitive case
  8. General case

Key Result

Theorem A

Let $a = (a(n))_{n=0}^\infty$ be a $k$-automatic sequence with effective alphabet size $r$ (cf. Definition def:eff-alph). Then for each $d \in \mathbb{N}$ we have

Theorems & Definitions (26)

  • Definition 1.1
  • Example 1.2
  • Theorem A
  • Remark 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • ...and 16 more