An Improved Bound for an Extension of Fine and Wilf's Theorem and Its Optimality

Considering two DNA molecules which are Watson-Crick (WK) complementary to each other “equivalent” with respect to the information they encode enables us to extend the classical notions of repetition, period, and power. WK-complementarity has been modelled mathematically by an antimorphic involution t, i.e., a function t such that t(xy) = t(y)t(x) for any x, y ∞ Sa, and t 2 is the identity. The WK-complementarity being thus modelled, any word which is a repetition of u and t(u) such as uu, ut(u)u, and ut(u)t(u)t(u) can be regarded repetitive in this sense, and hence, called a t-power of u. Taking the notion of t-power into account, the Fine and Wilf’s theorem was extended as “given an antimorphic involution t and words u, v, if a t-power of u and a t-power of v have a common prefix of length at least b(|u|, |v|) = 2|u| + |v| - gcd(|u|, |v|), then u and v are t-powers of a same word.” In this paper, we obtain an improved bound b′(|u|, |v|) = b(|u|, |v|) - [gcd(|u|, |v|)/2]. Then we show all the cases when this bound is optimal by providing all the pairs of words (u, v) such that they are not t-powers of a same word, but one can construct a t-power of u and a t-power of v whose maximal common prefix is of length equal to b′(|u|, |v|) − 1. Furthermore, we characterize such words in terms of Sturmian words.