On Brlek-Reutenauer conjecture

TL;DR

This paper proves the Brlek-Reutenauer conjecture for uniformly recurrent infinite words, linking their defect to complexities, and discusses related results and open problems to advance understanding of the conjecture.

Contribution

The paper extends the proof of the conjecture from periodic to uniformly recurrent words and summarizes related results and open problems.

Findings

Proved the conjecture for uniformly recurrent words.

Connected defect and complexity measures in infinite words.

Outlined open problems for full conjecture proof.

Abstract

Brlek and Reutenauer conjectured that any infinite word u with language closed under reversal satisfies the equality 2D(u)=\sum_{n=0}^{\infty} T(n) in which D(u) denotes the defect of u and T(n) denotes C(n+1)-C(n)+2-P(n+1)-P(n), where C and P are the factor and palindromic complexity of u, respectively. Brlek and Reutenauer verified their conjecture for periodic infinite words. We prove the conjecture for uniformly recurrent words. Moreover, we summarize results and some open problems related to defect, which may be useful for the proof of Brlek-Reutenauer Conjecture in full generality.