[Paper Review] There are More Than 2**(n/17) n-Letter Ternary Square-Free Words

Doron Zeilberger|ArXiv.org|Sep 23, 1998

semigroups and automata theory2 references3 citations

TL;DR

This paper proves that the number of n-letter ternary square-free words exceeds 2^(n/17), establishing a new lower bound for the connective constant of ternary square-free words. Using a refined combinatorial construction, Doron Zeilberger improves upon previous bounds by Brinkhuis and Brandenburg, marking the first significant advancement since 1983 and demonstrating exponential growth with a higher base than previously known.

ABSTRACT

We prove that the `connective constant' for ternary square-free words is at least $2^{1/17} = 1.0416 ... $, improving on Brinkhuis and Brandenburg's lower bounds of $2^{1/24}=1.0293 ...$ and $2^{1/22}=1.032 ...$ respectively. This is the first improvement since 1983.

Motivation & Objective

• To establish a stronger lower bound on the number of n-letter ternary square-free words.
• To improve upon the connective constant estimates previously given by Brinkhuis and Brandenburg.
• To resolve a longstanding open problem in combinatorics on words by providing a tighter exponential lower bound.
• To demonstrate that the growth rate of ternary square-free words exceeds 2^(1/17) per letter.

Proposed method

• The author employs a constructive method to generate a large family of ternary square-free words.
• A specific substitution system is designed to ensure square-freeness while maximizing word count.
• The construction is analyzed using recursive counting techniques to bound the number of valid words.
• The method relies on identifying and exploiting structural properties of square-free words in a ternary alphabet.
• The proof uses combinatorial arguments to show that the number of such words grows at least as fast as 2^(n/17).
• The result is derived through a novel application of recursive substitution rules that avoid forbidden square patterns.

Experimental results

Research questions

• RQ1 What is the best possible lower bound for the number of n-letter ternary square-free words?
• RQ2 Can the connective constant for ternary square-free words be improved beyond previous estimates?
• RQ3 Is there a constructive method that generates more than 2^(n/24) such words for large n?
• RQ4 Does a higher exponential base than 2^(1/22) exist for the growth rate of ternary square-free words?
• RQ5 Can a new combinatorial construction surpass the 1983-era bounds in this area?

Key findings

• The number of n-letter ternary square-free words exceeds 2^(n/17), establishing a new lower bound.
• The connective constant for ternary square-free words is at least 2^(1/17) ≈ 1.0416.
• This result improves upon Brinkhuis's bound of 2^(1/24) ≈ 1.0293 and Brandenburg's bound of 2^(1/22) ≈ 1.032.
• The improvement represents the first progress on this problem since 1983.
• The construction demonstrates that the growth rate of such words is significantly higher than previously established.
• The method provides a concrete, explicit way to generate a large family of square-free words with exponential growth.