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New examples of words for which binomial complexities and subword complexity coincide

Léo Vivion

Abstract

The complexity of an infinite word can be measured in several ways, the two most common measures being the subword complexity and the abelian complexity. In 2015, Rigo and Salimov introduced a family of intermediate complexities indexed by $k\in\mathbb{N}_{>0}$: the $k$-binomial complexities. These complexities scale up from the abelian complexity, with which the $1$-binomial complexity coincides, to the subword complexity, to which they converge pointwise as $k$ tends to $+\infty$. In this article, we provide four classes of $d$-ary infinite words -- namely, $d$-ary $1$-balanced words, words with subword complexity $n\in\mathbb{N}_{>0}\mapsto n+(d-1)$ (which form a subclass of the so-called quasi-Sturmian words), hypercubic billiard words, and words constructed by repeated Sturmian colorings -- for which this scale ``collapses'', that is, all $k$-binomial complexities, for $k\geq 2$, coincide with the subword complexity. This work generalizes a result of Rigo and Salimov, established in their seminal paper from 2015, which asserts that the $k$-binomial complexity of any Sturmian word coincides with its subword complexity whenever $k\geq 2$.

New examples of words for which binomial complexities and subword complexity coincide

Abstract

The complexity of an infinite word can be measured in several ways, the two most common measures being the subword complexity and the abelian complexity. In 2015, Rigo and Salimov introduced a family of intermediate complexities indexed by : the -binomial complexities. These complexities scale up from the abelian complexity, with which the -binomial complexity coincides, to the subword complexity, to which they converge pointwise as tends to . In this article, we provide four classes of -ary infinite words -- namely, -ary -balanced words, words with subword complexity (which form a subclass of the so-called quasi-Sturmian words), hypercubic billiard words, and words constructed by repeated Sturmian colorings -- for which this scale ``collapses'', that is, all -binomial complexities, for , coincide with the subword complexity. This work generalizes a result of Rigo and Salimov, established in their seminal paper from 2015, which asserts that the -binomial complexity of any Sturmian word coincides with its subword complexity whenever .

Paper Structure

This paper contains 14 sections, 15 theorems, 35 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and notation
  3. Basics and complexities
  4. Classes of words
  5. Comparison between the four classes of words in Theorem \ref{['th:main']}.
  6. Main results and proof strategies
  7. Binary $1$-balanced words
  8. Words with binary $1$-balanced projections
  9. Stability by coloring
  10. Proof of Proposition \ref{['prop:binary']}
  11. Proof of Lemma \ref{['lemma:projections_kbin']} and of Theorem \ref{['th:main']}, assertions (i)--(iii)
  12. Proof of Lemma \ref{['lemma:stability']} and Theorem \ref{['th:main']}, assertion (iv)
  13. Open questions
  14. Acknowledgments

Key Result

Proposition 1

The $1$-binomial complexity of an infinite word $w$ coincides with its subword complexity if and only if there exist $d$ distinct letters $a_1,\ldots,a_d$ and $(d-1)$ positive integers $k_1,\ldots,k_{d-1}$ such that where $a_d^\omega$ denotes the constant infinite word $a_da_da_da_d\ldots$.

Figures (1)

  • Figure 1: The ball, initially located in $x$ with a momentum $\theta$, generates the infinite word $w=1211212112...$

Theorems & Definitions (30)

  • Proposition 1: RSW24 Remark 7.1
  • Proposition 2: LRR20 Theorem 29
  • Conjecture 3: Lejeune, Rigo, Rosenfeld
  • Theorem 1
  • Remark 4
  • Remark 5
  • Proposition 6
  • Lemma 7
  • Lemma 8
  • proof : Proof of Lemma \ref{['lemma:binary_projections']}
  • ...and 20 more