The three squares lemma revisited

A recent paper Fan et al. (2006) [10] showed that the occurrence of two squares at the same position in a string, together with the occurrence of a third near by, is possible only in very special circumstances, represented by 14 well-defined cases. Similar results were published in Simpson (2007) [19]. In this paper we begin the process of extending this research in two ways: first, by proving a “two squares” lemma for a case not considered in Fan et al. (2006) [10]; second, by showing that in other cases, when three squares occur, more precise results — a breakdown into highly periodic substrings easily recognized in a left-to-right scan of the string — can be obtained with weaker assumptions. The motivation for this research is, first, to show that the maximum number of runs (maximal periodicities) in a string is at most n; second, and more important, to provide a combinatorial basis for a new generation of algorithms that directly compute repetitions in strings without elaborate preprocessing. Based on extensive computation, we present conjectures that describe the combinatorial behavior in all 14 of the subcases that arise. We then prove the correctness of seven of these conjectures. Along the way we establish a new combinatorial lemma characterizing strings of which two rotations have the same period.