Balances of $m$-bonacci words

TL;DR

This paper investigates the balance properties of $m$-bonacci words, establishing bounds on their letter frequency differences and improving known constants for small values of m through computational methods.

Contribution

It provides new bounds on the balance constants $c^{(m)}$ for $m$-bonacci words, including a general linear bound and improved specific values for small m.

Findings

The $m$-bonacci word is $(loor{\kappa m} + 12)$-balanced with $\kappa extasciitilde 0.58$.

For $m extless= 12$, the balance constant $c^{(m)}$ is improved to $\lceil (m+1)/2 ceil$.

The bounds are derived using a combination of theoretical analysis and computational verification.

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 $ \varphi_m: 0\to 01, 1\to 02, ..., (m-2)\to0(m-1), and (m-1)\to0$. 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 $m\geq2$. We show that the $m$-bonacci word is $(\lfloor \kappa m \rfloor +12)$-balanced, where $\kappa \approx 0.58$. For $m\leq 12$, we improve the constant $c^{(m)}$ by a computer numerical calculation to the value $\lceil\frac{m+1}{2}\rceil$.