On Combinatorial Properties of Greedy Wasserstein Minimization

We discuss a phenomenon where Optimal Transport leads to a remarkable amount of combinatorial regularity. Consider infinite sequences $(x_k)_{k=1}^{\infty}$ in $[0,1]$ constructed in a greedy manner: given $x_1, \dots, x_n$, the new point $x_{n+1}$ is chosen so as to minimize the Wasserstein distance $W_2$ between the empirical measure of the $n+1$ points and the Lebesgue measure, $$x_{n+1} = \arg\min_x ~W_2\left( \frac{1}{n+1} \sum_{k=1}^{n} \delta_{x_k} + \frac{\delta_{x}}{n+1}, dx\right).$$ This leads to fascinating sequences (for example: $x_{n+1} = (2k+1)/(2n+2)$ for some $k \in \mathbb{Z}$) which coincide with sequences recently introduced by Ralph Kritzinger in a different setting. Numerically, the regularity of these sequences rival the best known constructions from Combinatorics or Number Theory. We prove a regularity result below the square root barrier.