Ad-step approach to the maximum number of distinct squares and runs in strings

How many double squares can a string contain?

“…U [1] U [2] Thus, u is a prefix of v 1 U and u = v 1 k v 2 for some prefix v 2 of v 1 and k ≥ 1. Let u 1 be the primitive root of v 1 .…”

“…Let v 1 denote the overlap of U [1] with u [2] ; that is, u = v 1 v 1 for some v 1 and U = uv 1 , see the diagram below.…”

“…Thus x [1] refers to the first symbol of a string x, x [2] to the second etc. We use .. as a range symbol, thus x = x[1..n] is a string of length n, and x[i..j] refers to the substring, also often called factor, starting at position i and ending at position j.…”

“…For instance, u 1 = aabb, u 2 = aa, and p = 2 gives u 1 p u 2 = aabbaabbaa, and v 1 = aabbaabba, v 2 = a, and q = 1 gives v 1 q v 2 = aabbaabbaa; that is, u 1 p u 2 = v 1 q v 2 . As we often need to refer to the various occurrences of the same factor, we use a special subscript [1], [2], etc to distinguish them. For instance, u [1] may refer to the first occurrence of u in u 3 , while u [2] would refer to the second occurrence, etc.…”

“…Their proof relies on a lemma by Crochemore and Rytter [1] describing the relationship among the sizes of three primitively rooted squares starting at the same position. Not using Crochemore and Rytter's Lemma, Ilie [6] provided an alternative proof of Fraenkel and Simpson's result before presenting in [7] an asymptotic upper bound of 2n−Θ(log n) for sufficiently large n. A d-step approach to this problem introducing the size d of the alphabet as a parameter in addition to the length n of the string was proposed in [2]. Considering the maximum number σ d (n) of distinct primitively rooted squares over all strings of length n with exactly d distinct symbols, it is conjectured there that σ d (n) ≤ n−d.…”