Abstract
We consider repetitions in words and solve a longstanding open problem about the relation between the period and the length of its longest unbordered factor. A word u is called bordered if there exists a proper prefix that is also a suffix of u, otherwise it is called unbordered. In 1979 Ehrenfeucht and Silberger raised the following problem: What is the maximum length of a word w, w.r.t. the length τ of its longest unbordered factor, still allowing that τ is shorter than the period π of w. We show that if w is longer than 7(τ− 1)/3 then τ = π which gives the optimal asymtotic bound.
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Holub, Š., Nowotka, D. (2009). The Ehrenfeucht-Silberger Problem. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds) Automata, Languages and Programming. ICALP 2009. Lecture Notes in Computer Science, vol 5555. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02927-1_45
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DOI: https://doi.org/10.1007/978-3-642-02927-1_45
Publisher Name: Springer, Berlin, Heidelberg
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