|Abstract of Paper|
by Arturo Carpi, Aldo de Luca
We introduce the notion of periodiclike word. There exist several equiva lent definitions of this notion. In particular, a word is periodiclike if its longest repeated prefix is not right special. We recall that a factor $u$ of a word $w$ is right (resp.\ left) special if there exist two distinct letters $a$ and $b$ such that $ua$ and $ub$ (resp.\ $au$ and $bu$) are both factors of $w$. The class of periodiclike words properly contains that of periodic words and the class of semiperiodic words introduced in a previous paper. A parameter of great interest in the combinatorics of a periodiclike word $w$ is the least nonnegative integer $R_w'$ such that any prefix of $w$ of length $\geq R_w'$ is not right special. Indeed, we prove that a word $w$ is periodiclike if and only if $w$ has a period not larger than $|w|-R_w'$. Moreover, we establish the following result, which can be viewed as an extension of the theorem of Fine and Wilf: if a word $w$ has two periods $p,q\leq|w|-R_w'$, then gcd$(p, q)$ is a period of $w$. A proper box of a word $w$ is any factor of the kind $asb$ with $a$ and $b$ letters and $s$ both right and left special. This notion was introduced in some previous papers dealing with uniqueness conditions for words based on their `short' factors. Here, we show that if two words have the same set of maximal proper boxes either they are both periodiclike or none of them is periodiclike. Moreover, two periodiclike words have the same set of maximal proper boxes if and only if they are rootconjugate, i.e., if they have the same minimal period $p$ and their prefixes of length $p$ are conjugate. Finally, we present some uniqueness results related to the maximal box theorem.