Balances of $m$-bonacci words

Abstract

The $m$-bonacci word is a generalization of the Fibonacci word to the $m$-letter alphabet $\mathcal{A}=0,...,m-1$. It is the unique fixed point of the Pisot--type substitution ${\phi }_m:0\to 01,1\to 02,...,(m-2)\to 0(m-1),and(m-1)\to 0$. 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 $a\in \mathcal{A}$. 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 $m\ge 2$. We show that the $m$-bonacci word is $(\lfloor \kappa m\rfloor +12)$-balanced, where $\kappa \approx 0.58$. For $m\le 12$, we improve the constant $c^{(m)}$ by a computer numerical calculation to the value $\lceil \frac{m+1}{2}\rceil$.

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