@article {Brinda2014,
title = {Balances of m-bonacci words},
journal = {Fundamenta informaticae},
volume = {132},
number = {1},
year = {2014},
month = {jan},
pages = { 33-61},
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{\textendash}type substitution ϕm : 0 {\textrightarrow} 01, 1 {\textrightarrow} 02, . . . , (m - 2) {\textrightarrow} 0(m - 1), and (m - 1) {\textrightarrow} 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 ⌈ m+1 2 ⌉.},
doi = {10.3233/FI-2014-1011},
url = {http://dx.doi.org/10.3233/FI-2014-1031},
author = {Karel B{\v r}inda and Pelantov{\'a}, Edita and Turek, Ond{\v r}ej}
}