The Ehrenfeucht-Silberger Problem

Authors:
Štěpán Holub;Dirk Nowotka
Affiliations:
Department of Algebra, Charles University of Prague, Czech Republic;Institute for Formal Methods in Computer Science, Universität Stuttgart, Germany
Venue:
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Year:
2009

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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 *** of its longest unbordered factor, still allowing that *** is shorter than the period *** of w . We show that if w is longer than 7(*** *** 1)/3 then *** = *** which gives the optimal asymtotic bound.