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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.29600 |
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| _version_ | 1866918419690946560 |
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| author | Gwozdz, Maja |
| author_facet | Gwozdz, Maja |
| contents | 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 \[ λ_d(A_n)=\frac1N,\qquad A_n\subset B(x_n,6\sqrt d\,N^{-1/d})\qquad(1\le n\le N), \] and $λ_d\!\bigl(Q\setminus\bigcup_{n=1}^N A_n\bigr)=0$. 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 \[ W_\infty\!\left(\frac1N\sum_{n=1}^Nδ_{x_n},\,λ_d\right)\le 6\sqrt d\,N^{-1/d} \qquad(N\in\mathbb{N}). \] 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 $1\le p\le\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_29600 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Uniform optimal-order Wasserstein quantisation Gwozdz, Maja Classical Analysis and ODEs Number Theory 49Q22, 11K38, 11K31 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 \[ λ_d(A_n)=\frac1N,\qquad A_n\subset B(x_n,6\sqrt d\,N^{-1/d})\qquad(1\le n\le N), \] and $λ_d\!\bigl(Q\setminus\bigcup_{n=1}^N A_n\bigr)=0$. 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 \[ W_\infty\!\left(\frac1N\sum_{n=1}^Nδ_{x_n},\,λ_d\right)\le 6\sqrt d\,N^{-1/d} \qquad(N\in\mathbb{N}). \] 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 $1\le p\le\infty$. |
| title | Uniform optimal-order Wasserstein quantisation |
| topic | Classical Analysis and ODEs Number Theory 49Q22, 11K38, 11K31 |
| url | https://arxiv.org/abs/2603.29600 |