Following a recent paper of Anselmo et al., we consider $m \times n$ rectangular matrices formed from the Fibonacci word, and we show that their balance properties can be solved with a finite automaton. We also generalize the result to every Sturmian characteristic word corresponding to a quadratic irrational. Finally, we also examine the analogous question for the Tribonacci word and the Thue-Morse word.

翻译：基于Anselmo等人近期发表的一篇论文，我们研究了由斐波那契词构成的$m \times n$矩形矩阵，并证明其平衡性质可通过有限自动机求解。我们进一步将该结果推广至所有对应于二次无理数的Sturmian特征词。最后，我们还探讨了Tribonacci词与Thue-Morse词的类似问题。