环面与球面上数论点集在Wasserstein度量和L²偏差下的等分布
Equidistribution of number theoretic point sets in the torus and the sphere in Wasserstein metric and $L^2$ discrepancy
AI总结：
该研究确定了环面与球面上Kronecker序列、Fibonacci格点等算术点集在Wasserstein度量和L²偏差下的等分布速率，证明了两类度量间的严格不等式及对应最优性关联。
AI中文摘要：
我们确定了若干算术点构型在Wasserstein度量和关于球的L²偏差下的等分布精确速率。环面上的Kronecker序列与欧氏单位球面上的Fibonacci格点，在Wasserstein度量下的分布比随机点更均匀。环面上的模双曲线与模抛物线表现与随机点类似，不过其关于球的L²偏差渐近式中的常数与随机情形不同。我们还证明了环面或球面上任意概率测度的二次Wasserstein度量与关于球的L²偏差之间的严格不等式，特别地，我们表明任何在关于球的L²偏差下最优接近均匀测度的点集，在二次Wasserstein度量下也是最优的。
英文摘要：
We find the exact rate of equidistribution of several arithmetic point configurations in Wasserstein metric and $L^2$ discrepancy with respect to balls. Kronecker sequences in the torus and the Fibonacci lattice on the Euclidean unit sphere are shown to be more evenly distributed in Wasserstein metric than random points. Modular hyperbolas and modular parabolas in the torus behave like random points, although the constants in the asymptotics of the $L^2$ discrepancy with respect to balls differ from the random case. We also prove a sharp inequality between the quadratic Wasserstein metric and the $L^2$ discrepancy with respect to balls for an arbitrary probability measure on the torus or the sphere. In particular, we show that any point set that is optimally close to the uniform measure in $L^2$ discrepancy with respect to balls is also optimal in the quadratic Wasserstein metric.
