On square-free arithmetic progressions in infinite words
Let \(w\) be a word, and let \(w(i)\) stand for the \(i\)th letter of \(w\). The author shows that for each \(p\ge3\) there are infinite square-free ternary words \(w=w(1)w(2)w(3)\cdots\) such that the word \([w]_p=w(p)w(2p)w(3p)\cdots\) is square-free (Theorem 3). This result is tight since any square-free ternary word \(w\) with \([w]_2\) square-free has length at most 15. The construction in Theorem 3 relies on square-free cyclic shift morphisms found by \textit{J. Currie} [Theor. Comput. Sci. 482, 1--8 (2013; Zbl 1291.68302)] for \(p\ge23\) and on some cleverly selected square-free morphisms for each \(p=3,\dots,22\). The author concludes with three interesting open problems.
scientific article; zbMATH DE number 7050150
scientific article