The New Periodicity Lemma revisited

Some years ago a New Periodicity Lemma (NPL) was published, showing that the occurrence of two squares at any position i in a string necessarily precludes the occurrence of other squares of specified period in a specified neighbourhood of i. The proof of this lemma was complex, breaking down into 14 subcases, and required that the shorter of the two squares be regular. In this paper we significantly relax the conditions required by the NPL, in particular removing regularity altogether, and we establish a more precise result using a simpler proof based on lemmas that expose new combinatorial structures in a string, in particular a canonical factorization for any two squares that occur at the same position. We believe that these lemmas will also have application to other related problems such as the maximum number of runs and the maximum number of distinct squares.