Word of low complexity without uniform frequencies

TL;DR

This paper constructs a special type of infinite word that is both low in complexity and lacks uniform letter frequencies, demonstrating the optimality of a known theoretical bound.

Contribution

It presents a novel example of an infinite word with low complexity and non-uniform frequencies, confirming a key theoretical bound in combinatorics on words.

Findings

Existence of a low complexity, non-uniform frequency infinite word

Validation of Boshernitzan's bound as optimal

Insight into the structure of recurrent words

Abstract

In this paper, we construct a uniformely recurrent infinite word of low complexity without uniform frequencies of letters. This shows the optimality of a bound of Boshernitzan, which gives a sufficient condition for a uniformly recurrent infinite word to admit uniform frequencies.