Abstract of Paper

Periodic-like Words
by Arturo Carpi, Aldo de Luca

Abstract:

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.