The Ehrenfeucht–Silberger problem

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Abstract

We consider repetitions in words and solve a longstanding open problem about the relation between the period of a word and the length of its longest unbordered factor (where factor means uninterrupted subword). 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, such that τ is shorter than the period π of w. We show that, if w is of length 73τ or more, then τ=π which gives the optimal asymptotic bound.

Keywords

Combinatorics on words
Ehrenfeucht–Silberger problem
Periodicity
Unbordered words

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A preliminary version of this paper appeared in the proceedings of the ICALP 2009.

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The work on this article has been supported by the research project MSM 0021620839.