Some restrictions on periodicity in strings

Given a string x = x[1..n], a repetition of period p in x is a substring ur = x[i..i+rp−1], p = |u|, r ≥ 2, where neither u = x[i..i+p−1] nor x[i..i+(r+1)p−1] is a repetition. The maximum number of repetitions in any string x is well known to be Θ(n log n). A run or maximal periodicity of period p in x is a substring urt = x[i..i+rp+|t|−1] of x, where ur is a repetition, t a proper prefix of u, and no repetition of period p begins at position i−1 of x or ends at position i+rp+|t|. In 2000 Kolpakov & Kucherov showed that the maximum number ρ(n) of runs in any string x is O(n), but their proof was nonconstructive and provided no specific constant of proportionality. At the same time, they presented experimental data strongly suggesting that ρ(n) < n. that the maximum any string x again encourages the belief that in fact σ(n) < n. Recently, Fan et al.(“A new periodicity lemma”, Sixteenth Annual Symp. Combin. Pattern Matching, 2005) took a first step toward proving these conjectures, by presenting results that establish limitations on the number of squares of a specified range of periods that can occur over a specified range of positions in x. In this paper, we further tighten these restrictions by showing how the existence of two squares u and v (v longer than u) at the same position i in x limits the occurrence of smaller squares with period w ∈ (|v| − |u|, |u|) in the neighborhood around i