Cubefree words with many squares

TL;DR

This paper constructs infinite binary words that avoid cubes yet contain exponentially many distinct squares, and demonstrates the existence of cubefree binary squares of specific lengths, advancing understanding of pattern avoidance in words.

Contribution

It introduces new constructions of cubefree binary words with many squares and proves the existence of cubefree binary squares of certain lengths.

Findings

Infinite cubefree binary words with exponentially many squares

Existence of cubefree binary squares of length 2n for all n

Advances pattern avoidance in combinatorics on words

Abstract

We construct infinite cubefree binary words containing exponentially many distinct squares of length n. We also show that for every positive integer n, there is a cubefree binary square of length 2n.