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.
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