Initial nonrepetitive complexity of regular episturmian words and their Diophantine exponents

TL;DR

This paper develops a method to evaluate the initial nonrepetitive complexity of regular episturmian words, linking it to their Diophantine exponents and providing new insights into the irrationality of associated real numbers.

Contribution

It introduces a theory of generalized Ostrowski numeration systems for regular episturmian words and relates their complexity to Diophantine exponents, extending prior Sturmian word results.

Findings

Diophantine exponent is finite iff directive word has bounded partial quotients

Exponent exceeds 2 if partial quotients are eventually at least 3

Identifies uncountably many transcendental numbers with irrationality exponent > 2

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 $\xi_x$ having $x$ as a fractional part. The Diophantine exponent of $x$ is a lower bound for the irrationality exponent of $\xi_x$. 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.