Balances of m-bonacci Words

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 $\lceil {m+1 \over 2} \rceil$ .