Affiliations: FNSPE Czech Technical University in Prague, Trojanova 13, 120 00 Praha 2, Czech Republic. [email protected]; [email protected] | Laboratory of Physics, Kochi University of Technology, Tosa Yamada, Kochi 782-8502, Japan. [email protected]
Note: [] We acknowledge financial support by the Czech Science Foundation grant GAČR 201/09/0584, by the Grant Agency of the Czech Technical University in Prague, grant SGS11/162/OHK4/3T/14, and by the Foundation “Nadání Josefa, Marie a Zdeňka Hlávkových”. Address for correspondence: FNSPE Czech Technical University in Prague, Trojanova 13, 120 00 Praha 2, Czech Republic
Abstract: The m-bonacci word is a generalization of the Fibonacci word to the m-letter alphabet 𝒜 = {0, ... ,m − 1}. It is the unique fixed point of the Pisot–type substitution �m : 0 → 01, 1 → 02, ... , (m − 2) → 0(m − 1), and (m − 1) → 0. A result of Adamczewski implies the existence of constants c(m) such that the m-bonacci word is c(m)-balanced, i.e., numbers of letter a occurring in two factors of the same length differ at most by c(m) for any letter a ∈ 𝒜. The constants c(m) have been already determined for m = 2 and m = 3. In this paper we study the bounds c(m) for a general m ≥ 2. We show that the m-bonacci word is ($\lfloor; \kappa m \rfloor + 12$)-balanced, where κ ≈ 0.58. For m ≤ 12, we improve the constant c(m) by a computer numerical calculation to the value $\lceil {m+1 \over 2} \rceil$.
