On the irrationality exponent of real numbers with low complexity expansion

TL;DR

This paper investigates how the complexity of a real number’s base-$b$ expansion relates to its irrationality exponent, showing that numbers with typical irrationality exponent 2 have a certain minimal subword complexity growth.

Contribution

It establishes a lower bound on the subword complexity growth rate for real numbers with irrationality exponent 2, linking combinatorial properties of expansions to Diophantine approximation.

Findings

For almost all real numbers, the subword complexity growth rate is at least 4/3 times the length.

The proof involves analyzing Rauzy graphs of low complexity infinite words.

A lower bound on the subword complexity ratio is derived for numbers with irrationality exponent 2.

Abstract

Let $\xi$ be a real number and $b \ge 2$ an integer. We study the relationship between the irrationality exponent of $\xi$ and the subword complexity $p(n, \mathbf{x})$ of the $b$-ary expansion $\mathbf{x}$ of $\xi$, 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 $\xi$ is equal to $2$, which is the case for almost all real numbers $\xi$, 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.