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Initial nonrepetitive complexity of regular episturmian words and their Diophantine exponents

Jarkko Peltomäki

TL;DR

A theory of generalized Ostrowski numeration systems is developed and it is shown how to associate with each episturmian word a unique sequence of numbers written in this numeration system and yields nontrivial lower bounds for the irrationality exponents of real numbers whose fractional parts are regular epistURmian words.

Abstract

Regular episturmian words are episturmian words whose directive words have a regular and restricted form making them behave more like Sturmian words than general episturmian words. We present a method to evaluate the initial nonrepetitive complexity of regular episturmian words extending the work of Wojcik on Sturmian words. For this, we develop a theory of generalized Ostrowski numeration systems and show how to associate with each episturmian word a unique sequence of numbers written in this numeration system. The description of the initial nonrepetitive complexity allows us to obtain novel results on the Diophantine exponents of regular episturmian words. We prove that the Diophantine exponent of a regular episturmian word is finite if and only if its directive word has bounded partial quotients. Moreover, we prove that the Diophantine exponent of a regular episturmian word is strictly greater than $2$ if the sequence of partial quotients is eventually at least $3$. Given an infinite word $x$ over an integer alphabet, we may consider a real number $ξ_x$ having $x$ as a fractional part. The Diophantine exponent of $x$ is a lower bound for the irrationality exponent of $ξ_x$. Our results thus yield nontrivial lower bounds for the irrationality exponents of real numbers whose fractional parts are regular episturmian words. As a consequence, we identify a new uncountable class of transcendental numbers whose irrationality exponents are strictly greater than $2$. This class contains an uncountable subclass of Liouville numbers.

Initial nonrepetitive complexity of regular episturmian words and their Diophantine exponents

TL;DR

A theory of generalized Ostrowski numeration systems is developed and it is shown how to associate with each episturmian word a unique sequence of numbers written in this numeration system and yields nontrivial lower bounds for the irrationality exponents of real numbers whose fractional parts are regular epistURmian words.

Abstract

Regular episturmian words are episturmian words whose directive words have a regular and restricted form making them behave more like Sturmian words than general episturmian words. We present a method to evaluate the initial nonrepetitive complexity of regular episturmian words extending the work of Wojcik on Sturmian words. For this, we develop a theory of generalized Ostrowski numeration systems and show how to associate with each episturmian word a unique sequence of numbers written in this numeration system. The description of the initial nonrepetitive complexity allows us to obtain novel results on the Diophantine exponents of regular episturmian words. We prove that the Diophantine exponent of a regular episturmian word is finite if and only if its directive word has bounded partial quotients. Moreover, we prove that the Diophantine exponent of a regular episturmian word is strictly greater than if the sequence of partial quotients is eventually at least . Given an infinite word over an integer alphabet, we may consider a real number having as a fractional part. The Diophantine exponent of is a lower bound for the irrationality exponent of . Our results thus yield nontrivial lower bounds for the irrationality exponents of real numbers whose fractional parts are regular episturmian words. As a consequence, we identify a new uncountable class of transcendental numbers whose irrationality exponents are strictly greater than . This class contains an uncountable subclass of Liouville numbers.

Paper Structure

This paper contains 17 sections, 49 theorems, 165 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Episturmian Words
  3. Initial Nonrepetitive Complexity
  4. Diophantine Exponents and Main Results
  5. Preliminaries from Combinatorics on Words
  6. Episturmian Words and Generalized Ostrowski Numeration Systems
  7. The Intercept of an Episturmian Word
  8. Regular Episturmian Words
  9. Generalized Ostrowski Numeration Systems
  10. Auxiliary Results on Generalized Standard Words
  11. Rauzy Graphs of Episturmian Words
  12. Initial Nonrepetitive Complexity of Regular Episturmian Words
  13. Diophantine Exponents
  14. Diophantine Exponents of Regular Episturmian Words
  15. Diophantine Exponents of $d$-bonacci Words
  16. ...and 2 more sections

Key Result

theorem 1.2

Let $\mathbf{t}$ be a regular episturmian word of period $d$ with directive word $\Delta = x_1^{a_1} x_2^{a_2} \dotsm$. If $d = 2$ or $\limsup_k a_k \geq 3$, then $\mu(\xi_{ \mathbf{t},b}) > 2$.

Figures (2)

  • Figure 1: The Rauzy graph of an episturmian word. The left special factor corresponds to the vertex $\ell$ and the right special to the vertex $r$. The directed path from $\ell$ to $r$ is the central path.
  • Figure 2: Plots of the initial nonrepetitive complexity of the episturmian words with directive word $(001122)^\omega$ having intercepts $0^\omega$ (dark gray), $(01)^\omega$ (dashed light gray), and $1^\omega$ (black). The major ticks $1$, $5$, $17$, $51$, $147$ on the $x$-axis are the endpoints of the intervals $\mathcal{I}_{k}$, and the minor ticks are the endpoints of the intervals $\mathcal{I}_{k,0}$.

Theorems & Definitions (105)

  • definition 1.1
  • theorem 1.2
  • theorem 1.3
  • Lemma 3.1
  • proof
  • definition 3.2
  • Lemma 3.3
  • proof
  • definition 3.4
  • Proposition 3.5
  • ...and 95 more