Petrova–Shur's conjecture on the Abelian repetition threshold

For a finite alphabet of size $k$, let ${\sf ART}(k)$ be the infimum of the real numbers $\alpha$ for which there exists an infinite $k$-ary Abelian $\alpha$-power-free word. The paper studies these languages using randomized depth-first searches in their prefix trees.

Petrova–Shur's conjecture. The Abelian repetition thresholds satisfy

$${\sf ART}(2)>\frac{11}{3};\qquad 2<{\sf ART}(3)\le \frac{5}{2};\qquad {\sf ART}(4)>\frac{9}{5};\qquad {\sf ART}(5)=\frac{3}{2};$$ $$\frac{4}{3}<{sf ART}(6)<\frac{3}{2};\qquad {\sf ART}(k)=\frac{k-3}{k-4}\quad\text{for }k\ge 7.$$

These claims are based on experimental random walks and are intended to describe the observed finiteness or infiniteness of Abelian power-free languages. The authors prove the conjectured inequalities for $k=6,7,8,9,10$; the remaining assertions are experimental conjectures in the paper.