  In 1962, R. C. Lyndon and M. P. Shutzenberger established that for any
positive integers r and s, any sequence of length at least r+s that is both
r-periodic and s-periodic is then (r,s)-periodic. Shortly thereafter (1965), N.
J. Fine and H. S. Wilf proved that for any positive integers r and s, if a is
an infinite seqeunce of period r and b is an infinite sequence of period s such
that a_i=b_i for all i with 1\le i\le r+s-(r,s), then a=b. This is equivalent
to the following result, which is commonly referred to as the Fine-Wilf
theorem: for any positive integers r and s, if w is a finite sequence that is
both r-periodic and s-periodic, and |w|\ge r+s-(r,s), then w is (r,s)-periodic.
The Fine-Wilf theorem was generalized to finite sequences with three periods by
M. G. Castelli, F. Mignosi, and A. Restivo, and in general by J. Justin, and
even more broadly by R. Tijdeman and L. Zamboni. They introduced functions f
and fw from the set of all sequences of nonnegative integers to the set of
positive integers, and they proved that for a sequence p=(p_1,p_2,...,p_n), a
finite sequence w with periods p_i, i=1,2,..., n and length at least fw(p) must
be (p)-periodic as well, and that there exists a sequence w of length fw(p)-1
that is p_i-periodic for all i, but not (p)-periodic. In this paper, we follow
ideas introduced by S. Constantinescu and L. Ilie to obtain an alternative
formulation of f and fw, and we establish important properties of f and fw,
obtaining in particular new upper and lower bounds for each. We also begin an
investigation of Fine-Wilf graphs for arbitrary finite sequences.
