Uniform optimal-order Wasserstein quantisation

Abstract

We address Steinerberger's Wasserstein transport problem on the cube $Q=[0,1]^d$. For every $d\ge2$, we consider a dyadic digital sequence $(x_n)\subset Q$ and prove that every prefix $\{x_1,\dots,x_N\}$ admits an exact equal-mass transport partition at the optimal scale. More precisely, for every $N\in\mathbb{N}$, there exist pairwise disjoint Borel sets $A_1,\dots,A_N\subset Q$ such that \[ \lambda_d(A_n)=\frac1N,\qquad A_n\subset B(x_n,6\sqrt d\,N^{-1/d})\qquad(1\le n\le N), \] and $\lambda_d\!\bigl(Q\setminus\bigcup_{n=1}^N A_n\bigr)=0$. In other terms, every prefix of the sequence supports an exact transport allocation of Lebesgue mass to its points with uniformly controlled radius $O(N^{-1/d})$. By an elementary partition criterion, this yields \[ W_\infty\!\left(\frac1N\sum_{n=1}^N\delta_{x_n},\,\lambda_d\right)\le 6\sqrt d\,N^{-1/d} \qquad(N\in\mathbb{N}). \] The bound holds for every $1\le p\le\infty$. The exponent $1/d$ is optimal, so it gives the sharp uniform prefix rate on the cube. The result settles Steinerberger's problem for all $d\ge1$ and all $1\le p\le\infty$.