2022/07/17 by Steinerberger, Stefan
#Classical Analysis and ODEs (math.CA) #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2207.08043
We discuss a phenomenon where Optimal Transport leads to a remarkable amount of combinatorial regularity. Consider infinite sequences (xk)k=1∞ in [0,1] constructed in a greedy manner: given x1, …, xn, the new point xn+1 is chosen so as to minimize the Wasserstein distance W2 between the empirical measure of the n+1 points and the Lebesgue measure, xn+1 = argminx ~W2( (1)/(n+1) ∑k=1n δxk + \fracδxn+1, dx). This leads to fascinating sequences (for example: xn+1 = (2k+1)/(2n+2) for some k ∈ 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.