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We establish asymptotic upper and lower bounds for the Wasserstein distance of any order p ≥ 1 between the empirical measure of a fractional Brownian motion on a flat torus and the uniform Lebesgue measure. Our inequalities reveal an interesting interaction between the Hurst index H and the dimension d of the state space, with a "phase-transition" in the rates when d = 2 + 1 / H , akin to the Ajtai–Komlós–Tusnády theorem for the optimal matching of i.i.d. points in two-dimensions. Our proof couples PDE's and probabilistic techniques, and also yields a similar result for discrete-time approximations of the process, as well as a lower bound for the same problem on R d. [ABSTRACT FROM AUTHOR]