  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.
