The Weak Circular Repetition Threshold Over Large Alphabets

The repetition threshold for words on $n$ letters, denoted \mbox{RT}(n), is the infimum of the set of all $r$ such that there are arbitrarily long $r$-free words over $n$ letters. A repetition threshold for circular words on $n$ letters can be defined in three natural ways, which gives rise to the weak, intermediate, and strong circular repetition thresholds for $n$ letters, denoted \mbox{CRT}_{\mbox{W}}(n), \mbox{CRT}_{\mbox{I}}(n), and \mbox{CRT}_{\mbox{S}}(n), respectively. Currie and the present authors conjectured that \mbox{CRT}_{\mbox{I}}(n)=\mbox{CRT}_{\mbox{W}}(n)=\mbox{RT}(n) for all $n\geq 4$. We prove that \mbox{CRT}_{\mbox{W}}(n)=\mbox{RT}(n) for all $n\geq 45$, which confirms a weak version of this conjecture for all but finitely many values of $n$.Comment: arXiv admin note: text overlap with arXiv:1911.0577

Branching densities of cube-free and square-free words

Binary cube-free language and ternary square-free language are two “canonical” represen-tatives of a wide class of languages defined by avoidance properties. Each of these two languages can be viewed as an infinite binary tree reflecting the prefix order of its elements. We study how “homogenious” these trees are, analysing the following parameter: the density of branching nodes along infinite paths. We present combinatorial results and an efficient search algorithm, which together allowed us to get the following numerical results for the cube-free language: the minimal density of branching points is between 3509/9120 ≈ 0.38476 and 13/29 ≈ 0.44828, and the maximal density is between 0.72 and 67/93 ≈ 0.72043. We also prove the lower bound 223/868 ≈ 0.25691 on the density of branching points in the tree of the ternary square-free language. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.This research was funded by Ministry of Science and Higher Education of the Russian Federation (Ural Mathematical Center project No. 075-02-2020-1537/1)