Square-Density Increasing Mappings
The square conjecture claims that the number of distinct squares, factors of the form xx, in a word is at most the length of the word. Being associated with it, it is also conjectured that binary words have the largest square density. That is, it is sufficient to solve the square conjecture for words over binary alphabet. We solve this subsidiary conjecture affirmatively, or more strongly, we prove the irrelevance of the alphabet size in solving the square conjecture, as long as the alphabet is not unary. The tools we employ are homomorphisms with which one can convert an arbitrary word into a word with strictly larger square density over an intended alphabet.
KeywordsAlphabet Size Letter Alphabet Unary Word Positive Rational Number Algebraic Framework
We appreciate fruitful discussions with James Currie at Christian-Albrechts-University of Kiel, with Hideo Bannai and Simon Puglisi at the University of Helsinki, and with Nataša Jonoska when she visited Aalto University.