The Ehrenfeucht-Silberger problem

Authors:
Štpán Holub;Dirk Nowotka
Affiliations:
Department of Algebra, Charles University of Prague, Sokolovska 83, 186 75 Praha 8, Czech Republic;Department of Computer Science, Christian-Albrechts-Universität zu Kiel, Christian-Albrechts-Platz 4, 24118 Kiel, Germany
Venue:
Journal of Combinatorial Theory Series A
Year:
2012

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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 @t of its longest unbordered factor, such that @t is shorter than the period @p of w. We show that, if w is of length 73@t or more, then @t=@p which gives the optimal asymptotic bound.