Periodic-like words, periodicity, and boxes

Abstract.

We introduce the notion of periodic-like word. It is a word whose longest repeated prefix is not right special. Some different characterizations of this concept are given. In particular, we show that a word w is periodic-like if and only if it has a period not larger than $|w|-R'_{w}$, where $R'_{w}$ is the least non-negative integer such that any prefix of w of length $\geq R'_{w}$ is not right special. We derive that if a word w has two periods $p,q\leq |w|-R'_{w}$, then also the greatest common divisor of p andq is a period ofw. This result is, in fact, an improvement of the theorem of Fine and Wilf. We also prove that the minimal period of a word w is equal to the sum of the minimal periods of its components in a suitable canonical decomposition in periodic-like subwords. Moreover, we characterize periodic-like words having the same set of proper boxes, in terms of the important notion of root-conjugacy. Finally, some new uniqueness conditions for words, related to the maximal box theorem are given.