A Note on Squares in Binary Words

08/10/2021

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by   Tero Harju, et al.

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We consider words over a binary alphabet. A word w is overlap-free if it does not have factors (blocks of consecutive letters) of the form uvuvu for nonempty u. Let M(w) denote the number of positions that are middle positions of squares in w. We show that for overlap-free binary words, 2M(w) ≤ |w|+3, and that there are infinitely many overlap-free binary words for which 2M(w)=|w|+3.