A new complexity function, repetitions in Sturmian words, and irrationality exponents of Sturmian numbers

TL;DR

This paper introduces a new complexity measure for infinite words, characterizes Sturmian words using it, and applies these results to establish optimal bounds on the irrationality exponents of Sturmian numbers, with implications for their digit expansions.

Contribution

It presents a novel complexity function based on second occurrence times, characterizes Sturmian words with this function, and derives optimal bounds on irrationality exponents of Sturmian numbers.

Findings

Characterization of Sturmian words via the new complexity function

Establishment of a best-possible result on repetitions in Sturmian words

Lower bounds for irrationality exponents of Sturmian numbers and applications to digit expansions

Abstract

We introduce and study a new complexity function in combinatorics on words, which takes into account the smallest second occurrence time of a factor of an infinite word. We characterize the eventually periodic words and the Sturmian words by means of this function. Then, we establish a new result on repetitions in Sturmian words and show that it is best possible. Let $b \ge 2$ be an integer. We deduce a lower bound for the irrationality exponent of real numbers whose sequence of $b$-ary digits is a Sturmian sequence over $\{0,1,\ldots, b-1\}$ and we prove that this lower bound is best possible. As an application, we derive some information on the $b$-ary expansion of $\log(1+\frac{1}{a})$,for any integer $a \ge 34$.