The Entropy of Square-Free Words

TL;DR

This paper investigates the entropy density of infinite square-free words over alphabets of at least three letters, providing bounds and an asymptotically accurate approximation formula.

Contribution

It establishes positive entropy density for such words and derives bounds and an approximate formula with rapid convergence.

Findings

Entropy density is strictly positive for infinite square-free words.

Derived lower and upper bounds for entropy density.

Presented an approximate formula that converges rapidly asymptotically.

Abstract

Finite alphabets of at least three letters permit the construction of square-free words of infinite length. We show that the entropy density is strictly positive and derive reasonable lower and upper bounds. Finally, we present an approximate formula which is asymptotically exact with rapid convergence in the number of letters.