Balanced Fibonacci word rectangles, and beyond
🤖 AI Summary
This paper investigates the balance property of $m imes n$ rectangular matrices formed from Fibonacci words, and generalizes the analysis to all Sturmian characteristic words associated with quadratic irrational slopes. To address this problem, we construct, for the first time, a finite automaton that efficiently decides the balance of rectangular arrays of any size derived from Fibonacci words. We further demonstrate that this automaton construction systematically extends to all Sturmian characteristic words whose slopes are quadratic irrationals. Our work establishes a novel connection between the combinatorial structure of Sturmian words and the decision power of finite automata. It provides the first automata-theoretic formal decision framework for the classical combinatorial language-theoretic property of balance. By bridging combinatorics on words, ergodic theory, and formal language theory, this result expands the interdisciplinary boundaries among these fields.
Application Category
📝 Abstract
Following a recent paper of Anselmo et al., we consider $m imes n$ 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.
Problem
Research questions and friction points this paper is trying to address.
Analyzing balance properties of Fibonacci word matrices
Developing finite automaton solutions for rectangular patterns
Extending results to quadratic irrational Sturmian words
Innovation
Methods, ideas, or system contributions that make the work stand out.
Balanced Fibonacci word rectangle properties
Solved using finite automaton techniques
Generalized to Sturmian characteristic words
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