Fine-Wilf图与推广的Fine-Wilf定理
Fine-Wilf graphs and the generalized Fine-Wilf theorem
摘要：
1962年，R. C. Lyndon和M. P. Shutzenberger建立了对于任何正整数r和s，任何长度至少为r+s的序列，如果同时是r-周期和s-周期的，则是（r，s）-周期的。1965年，N。Fine和H. S. Wilf证明了对任意正整数r和s，若a是周期为r的无穷序列，b是周期为s的无穷序列，使得对所有i有1个\le i \le r+s-（r，s），a_i=b_i，则a=b.这等价于下面的结果，通常被称为Fine-Wilf定理：对于任何正整数r和s，如果w是一个既r-周期又s-周期的有限序列，并且|w| \ge r+s-（r，s），则w是（r，s）-周期的。M. Fine-Wilf定理推广到三周期有限序列。G.卡斯泰利角Mignosi和A. Restivo，并在一般的J.贾斯汀，甚至更广泛的R。Tijdeman和L.磨冰机他们把函数f和fw从所有非负整数序列的集合引入到正整数的集合中，并证明了对于一个序列p=（p_1，p_2，. p_n），周期为p_i的有限序列w，i= 1，2，.，n且长度至少为fw（p）的序列也必须是（p）周期的，并且存在长度为fw（p）-1的序列w，该序列对于所有i都是p_i周期的，但不是（p）周期的。在本文中，我们遵循S。Constantinescu和L. Ilie得到一个替代制定的f和fw，我们建立了重要的性质f和fw，特别是获得新的上限和下界为每个。我们也开始调查的Fine-Wilf图的任意有限序列。
英文摘要：
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.
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