Prefixes of the Fibonacci word

TL;DR

This paper provides an explicit proof of a theorem about the repetitive structure of the Fibonacci word's prefixes, using automata theory and logic to deepen understanding of its pattern complexity.

Contribution

It introduces an explicit version of a known theorem on Fibonacci word prefixes' repetitive structure, employing automata and logic tools.

Findings

Explicit bounds for prefix repetitions in Fibonacci words

Enhanced understanding of Fibonacci word repetitive structure

Application of automata theory to combinatorics on words

Abstract

Mignosi, Restivo, and Salemi (1998) proved that for all $\epsilon > 0$ there exists an integer $N$ such that all prefixes of the Fibonacci word of length $\geq N$ contain a suffix of exponent $\alpha^2-\epsilon$, where $\alpha = (1+\sqrt{5})/2$ is the golden ratio. In this note we show how to prove an explicit version of this theorem with tools from automata theory and logic. Along the way we gain a better understanding of the repetitive structure of the Fibonacci word.