Balanced Fibonacci word rectangles, and beyond
This work addresses a theoretical problem in combinatorics and formal language theory, focusing on word properties and automata, and is incremental as it builds on prior research by Anselmo et al.
The paper tackled the problem of analyzing balance properties of rectangular matrices formed from the Fibonacci word, showing that these properties can be solved using a finite automaton, and extended this result to Sturmian characteristic words for quadratic irrationals and examined similar questions for the Tribonacci and Thue-Morse words.
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.