On the Tree of Ternary Square-Free Words

Branching Densities of Cube-Free and Square-Free Words

“…We establish the lower bound 3509/9120 ≈ 0.38476 on the branching density in the prefix tree of CF (Theorem 3) and the lower bound 223/868 ≈ 0.25691 on the branching density in the prefix tree of SF (Theorem 4), significantly improving the bounds from [19,23]; • We construct infinite paths in the prefix tree of CF with the branching density as small as 13/29 ≈ 0.44828 (Theorem 5); •…”

“…In the same way, we can get the lower bound for the ternary square-free language SF. From [19] (Lemma 5), we have the upper bound 2 h for the density of positions fixed by squares of periods ≥ h. Lemmas 4 and 5, and Corollary 1 have direct analogs for ternary square-free words; Proposition 1 and the algorithm inside remain valid for any automaton having, at most, two outgoing edges for each state. Running the algorithm for the regular approximations of SF up to i = 30, we obtained the correspondent numbers d i .…”

“…Note that the prefix tree of SF is binary in spite of the ternary alphabet, because a square-free word has no factors of the form aa. For the prefix tree of SF, it is known that (a) finite subtrees of arbitrary depth exist and can be built efficiently [18], (b) in any infinite path, the fraction of nodes with two children is at least 2/9 [19], and (c) if a node of depth n has a single descendant of depth n + m, then m = O(log n) [19]. If we take the tree consisting of all infinite branches of the original prefix tree, then the analog of (c) with the bound m = O(n 2/3 ) is known [20,21].…”

“…There is a chance that the new bound can be slightly improved if more computational resources will be used. We also use the same search algorithm to improve the bound for the square-free language; the combinatorial part, presented in [19], is much simpler than for the cube-free case, and we see no way to improve it.…”

Transition Property for Cube-Free Words

“…, p k } can belong to the range [p..2p] for some fixed p ≥ 2. This is an analog of [13,Lemmas 4,5] and [14,Lemma 9]. Let i 0 < i 1 < · · · < i s be the list of all positions such that the periodic suffix of u·w [1..i j ] has the period from the range [p..2p]; let q 0 , .…”