A regularity lemma and twins in words

Authors:
Maria Axenovich;Yury Person;Svetlana Puzynina
Affiliations:
Iowa State University, Ames, IA, USA and Karlsruher Institut für Technologie, Karlsruhe, Germany;Freie Universität Berlin, Institut für Mathematik, Berlin, Germany;University of Turku, Turku, Finland and Sobolev Institute of Mathematics, Novosibirsk, Russia
Venue:
Journal of Combinatorial Theory Series A
Year:
2013

Citing 7
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Abstract

For a word S, let f(S) be the largest integer m such that there are two disjoint identical (scattered) subwords of length m. Let f(n,@S)=min{f(S):S is of length n, over alphabet @S}. Here, it is shown that2f(n,{0,1})=n-o(n) using the regularity lemma for words. In other words, any binary word of length n can be split into two identical subwords (referred to as twins) and, perhaps, a remaining subword of length o(n). A similar result is proven for k identical subwords of a word over an alphabet with at most k letters.