Periods, Partial Words, and a Result of Guibas and Odlyzko 
F. BlanchetSadri and Brian Shirey 
A well known and unexpected result of Guibas and
Odlyzko states that the set of all periods of a word is independent of the
alphabet size (alphabets with one symbol are excluded here).
More specifically, for every word u,
there exists a word v over the
alphabet {0,1} such that u and
v have the same length and the same set of periods. Recently,
BlanchetSadri and Chriscoe extended this fundamental result to words with one
"do not know" symbol also called partial words with one hole.
They showed that for every partial word u with one hole,
there exists a partial word v with at most one hole over the alphabet {0,1}
such that u and v
have the same length, the same set of periods, the same set of weak
periods, and H(v) Ì H(u), where H(u)
(respectively, H(v)) denotes the set of holes of u
(respectively, v). In this paper, we extend
this result further to a large class of partial words.
Given a partial word u belonging to that class,
our proof provides an algorithm to compute a partial word v
over {0,1} sharing the same length and same sets of periods and weak periods as u,
and satisfying H(v) Ì H(u).
Keywords: Words, partial words, periods, and weak periods. 

