Uniform optimal-order Wasserstein quantisation

We address Steinerberger’s Wasserstein transport problem on the cube $Q=[0,1{]}^d$. For every $d\ge 2$, 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 $${\lambda }_d(A_n)=\frac{1}{N},A_n\subset B(x_n,6\sqrt{d}{N}^{-1\mathrm{/}d})(1\le n\le N),$$ and ${\lambda }_d\text{\hspace{0.17em}⁣}(Q\setminus {\bigcup }_{n=1}^N{A}_n)=0$. In other terms, every prefix of the sequence supports an exact transport allocation of Lebesgue mass to its points with uniformly controlled radius $O(N^{-1\mathrm{/}d})$. By an elementary partition criterion, this yields $$W_{\mathrm{\infty }}\text{\hspace{0.17em}⁣}(\frac{1}{N}\sum _{n=1}^N{\delta }_{x_n},{\lambda }_d)\le 6\sqrt{d}{N}^{-1\mathrm{/}d}(N\in \mathbb{N})\mathrm{.}$$ The bound holds for every $1\le p\le \mathrm{\infty }$. The exponent $1\mathrm{/}d$ is optimal, so it gives the sharp uniform prefix rate on the cube. The result settles Steinerberger’s problem for all $d\ge 1$ and all $1\le p\le \mathrm{\infty }$.