On repetitiveness measures of Thue-Morse words

Abstract

We show that the size γ(tn) of the smallest string attractor of the nth Thue-Morse word tn is 4 for any n ≥ 4, disproving the conjecture by Mantaci et al. [ICTCS 2019] that it is n. We also show that δ(tn) = 10 3+2 4−n for n ≥ 3, where δ(w) is the maximum over all k = 1, . . . , |w|, the number of distinct substrings of length k in w divided by k, which is a measure of repetitiveness recently studied by Kociumaka et al. [LATIN 2020]. Furthermore, we show that the number z(tn) of factors in the self-referencing Lempel-Ziv factorization of tn is exactly 2n.