On the irrationality exponent of real numbers with low complexity expansion

TL;DR

This work links the irrationality exponent $\mu(\xi)$ to the subword complexity $p(n, \xi, b)$ of the base-$b$ expansion by analyzing the Rauzy-graph structure of the associated infinite word. A full combinatorial framework is developed to study words of low complexity, including precise notions of left/right/bi-special words, recurrence, and reduced Rauzy graphs, with a focus on $\infty$-shape configurations. The main result shows that if $\mu(\xi)=2$, then $\limsup_{n\to\infty} p(n, \xi, b)/n \ge 4/3$, and provides a $\mu$-dependent lower bound for $\limsup$ when $\mu>2$, derived via recurrence-time arguments and the evolution of Rauzy graphs. This approach yields stronger limsup bounds than prior methods based on repetition measures, illuminating how combinatorial structure in low-complexity expansions constrains Diophantine approximation properties.

Abstract

Let $ξ$ be a real number and $b \ge 2$ an integer. We study the relationship between the irrationality exponent of $ξ$ and the subword complexity $p(n, \mathbf{x})$ of the $b$-ary expansion $\mathbf{x}$ of $ξ$, where $p(n, \mathbf{x})$ counts the number of distinct blocks of length $n$ in $\mathbf{x}$, for $n \ge 1$. If the irrationality exponent of $ξ$ is equal to $2$, which is the case for almost all real numbers $ξ$, we show that the limit superior of the sequence $(p(n, \mathbf{x}) / n)_{n \ge 1}$ is at least equal to 4/3. The proof is based on a careful study of the evolution of the Rauzy graphs of infinite words of low complexity.

Figures (2)

• Figure 1: The Rauzy graph ${\mathcal{G}}_n({\mathbf x})$ with a bispecial word $w$ and two cycles $U$, $V$
• Figure 2: The reduced Rauzy graph $\mathcal{G}'_{n+k}({\mathbf x})$

Theorems & Definitions (43)

• Theorem 1.1: BuKi17
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• Definition 2.4
• Definition 2.5