Formal languages
Text algorithms
Handbook of formal languages, vol. 1
Fine and Wilf's theorem for three periods and a generalization of Sturmian words
Theoretical Computer Science
Generalised fine and Wilf's theorem for arbitrary number of periods
Theoretical Computer Science - Combinatorics on words
An Improved Bound for an Extension of Fine and Wilf’s Theorem and Its Optimality
Fundamenta Informaticae
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In 1965, Fine and Wilf proved the following theorem: if (fn)n≥0 and (gn)n≥0 are periodic sequences of real numbers, of period lengths h and k, respectively, and fn = gn for 0 ≤ n h + k - gcd(h,k), then fn = gn for all n ≥ 0. Furthermore, the constant h + k - gcd(h,k) is best possible. In this paper, we consider some variations on this theorem. In particular, we study the case where fn ≤ gn, instead of fn = gn. We also obtain generalizations to more than two periods.We apply our methods to a previously unsolved conjecture on iterated morphisms, the decreasing length conjecture: if h : Σ* → Σ* is a morphism with |Σ|= n, and w is a word with |w| h(w)| h2(w)| hk(w)|, then k ≤ n.