We present a short and elementary proof of the Ajtai–Komlós–Tusnády (AKT) optimal matching theorem in dimension 2 via Fourier analysis and a smoothing argument. The upper bound applies to more general families of samples, including dependent variables, of interest in the study of rates of convergence for empirical measures. Following the recent pde approach by L. Ambrosio, F. Stra and D. Trevisan, we also adapt a simple proof of the lower bound.
Research of S.B. was partially supported by NSF Grant DMS-1855575.
The authors thank the reviewer for relevant suggestions of improvements in the exposition, and for pointing out a technical issue in one proof.
"A simple Fourier analytic proof of the AKT optimal matching theorem." Ann. Appl. Probab. 31 (6) 2567 - 2584, December 2021. https://doi.org/10.1214/20-AAP1656