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 ϕm: 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 ([κm] + 12)-balanced, where κ ≈ 0.58. For m ≤ 12, we improve the constant c(m) by a computer numerical calculation to the value [wzór ].