Balancedness of Arnoux-Rauzy and Brun words
Vincent Delecroix (IMJ), Tom\'a\v{s} Hejda, Wolfgang Steiner (LIAFA)
TL;DR
This paper investigates the balancedness properties of Arnoux-Rauzy and Brun words generated by multi-dimensional continued fraction algorithms, showing that most are finitely balanced, with some exceptions.
Contribution
It establishes that almost all Brun and Arnoux-Rauzy words are finitely balanced, linking boundedness of partial quotients to balancedness, and provides counterexamples of unbalanced Brun words.
Findings
Most Brun and Arnoux-Rauzy words are finitely balanced.
Boundedness of strong partial quotients implies balancedness.
Counterexamples of unbalanced Brun words on 3 letters.
Abstract
We study balancedness properties of words given by the Arnoux-Rauzy and Brun multi-dimensional continued fraction algorithms. We show that almost all Brun words on 3 letters and Arnoux-Rauzy words over arbitrary alphabets are finitely balanced; in particular, boundedness of the strong partial quotients implies balancedness. On the other hand, we provide examples of unbalanced Brun words on 3 letters.
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