We discuss the classical problem of measuring the regularity of distribution of sets of $N$ points in $\mathbb{T}^d$. A recent line of investigation is to study the cost ($=$ mass $\times$ distance) necessary to move Dirac measures placed in these points to the uniform distribution. We show that Kronecker sequences satisfy optimal transport distance in $d \geq 3$ dimensions. This shows that for differentiable $f: \mathbb{T}^d \rightarrow \mathbb{R}$ and badly approximable vectors $\alpha \in \mathbb{R}^d$, we have $$ \ | \int_{\mathbb{T}^d} f(x) dx - \frac{1}{N} \sum_{k=1}^{N} f(k \alpha) \ | \leq c_{\alpha} \frac{ \| \nabla f\|^{(d-1)/d}_{L^{\infty}}\| \nabla f\|^{1/d}_{L^{2}} }{N^{1/d}}.$$ We note that the result is uniformly true for a sequence instead of a set. Simultaneously, it refines the classical integration error for Lipschitz functions, $\| \nabla f\|_{L^{\infty}} N^{-1/d}$. We obtain a similar improvement for numerical integration with respect to the regular grid. The main ingredient is an estimate involving Fourier coefficients of a measure; this allows for existing estimates to be conviently `recycled'. We present several open problems.

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关于经典序列和勒贝格测度之间的 Wasserstein 距离

我们讨论了测量 $\mathbb{T}^d$ 中 $N$ 点集合的分布规律的经典问题。最近的一项调查是研究将放置在这些点中的狄拉克测度移动到均匀分布所需的成本（$=$ 质量 $\times$ 距离）。我们表明 Kronecker 序列在 $d\geq 3$ 维度上满足最佳传输距离。这表明对于可微的 $f: \mathbb{T}^d \rightarrow \mathbb{R}$ 和非常近似的向量 $\alpha \in \mathbb{R}^d$，我们有 $$ \ | \int_{\mathbb{T}^d} f(x) dx - \frac{1}{N} \sum_{k=1}^{N} f(k \alpha) \ | \leq c_{\alpha} \frac{ \| \nabla f\|^{(d-1)/d}_{L^{\infty}}\| \nabla f\|^{1/d}_{L^{2}} }{N^{1/d}}.$$ 我们注意到结果对于序列而不是集合都是一致的。同时，它改进了 Lipschitz 函数的经典积分误差，$\| \nabla f\|_{L^{\infty}} N^{-1/d}$。我们获得了相对于规则网格的数值积分的类似改进。主要成分是涉及度量的傅立叶系数的估计；这样就可以方便地"回收"现有的估计数。我们提出了几个未解决的问题。