We describe a curious dynamical system that results in sequences of real numbers in $[0,1]$ with seemingly remarkable properties. Let the even function $f:\mathbb{T}\to \mathbb{R}$ satisfy $\widehat{f}\left(k\right)\ge c{\left|k\right|}^{-2}$ and define a sequence via $x_n=arg\underset{x}{min}\sum _{k=1}^{n-1}f(x-x_k).$ Such sequences ${\left(x_n\right)}_{n=1}^{\infty }$ seem to be astonishingly regularly distributed in various ways (satisfying favorable exponential sum estimates; every interval $J\subset [0,1]$ contains $\sim \left|J\right|n$ elements). We prove $W_2\left(\mu ,\nu \right)\le \frac{c}{\sqrt{n}},\text{where}\mu =\frac{1}{n}\sum _{k=1}^n{\delta }_{x_k}$ is the empirical distribution, $\nu =dx$ is the Lebesgue measure and $W_2(\mu ,\nu )$ is the 2-Wasserstein distance between these two. Much stronger results seem to be true and it is an interesting problem to understand this dynamical system better. We obtain optimal results in dimension $d\ge 3$: using $G(x,y)$ to denote the Green’s function of the Laplacian on a compact manifold, we show that $x_n=arg\underset{x\in M}{min}\sum _{k=1}^{n-1}G(x,x_k)\text{satisfies}{W}_2\left(\frac{1}{n}\sum _{k=1}^n{\delta }_{x_k},dx\right)\lesssim \frac{1}{n^{1∕d}}.$

我们描述了一个好奇的动力系统，该系统产生实数序列 $[0，\mathrm{1个}]$具有看似卓越的性能。让偶函数$F：\mathbb{Ť}\to \mathbb{[R}$ 满足 $\widehat{F}（ķ）\ge C{\left|ķ\right|}^{-2}$ 并通过定义一个序列 $X_{ñ}=精氨酸\underset{X}{分}\sum _{ķ=\mathrm{1个}}^{ñ-\mathrm{1个}}F（X-X_{ķ}）。$ 这样的序列 ${（X_{ñ}）}_{ñ=\mathrm{1个}}^{\infty }$ 似乎以各种方式惊人地规则分布（满足令人满意的指数和估计；每个间隔 $Ĵ\subset [0，\mathrm{1个}]$ 包含 $〜\left|Ĵ\right|ñ$元素）。我们证明${\mathrm{w\; ^}}_2\left(\mu ，\nu \right)\le \frac{C}{\sqrt{ñ}}，\text{哪里}\mu =\frac{\mathrm{1个}}{ñ}\sum _{ķ=\mathrm{1个}}^{ñ}{\delta }_{X_{ķ}}$ 是经验分布， $\nu =dX$ 是勒贝格的措施， ${\mathrm{w\; ^}}_2（\mu ，\nu ）$是两者之间的2-Wasserstein距离。似乎更强的结果是正确的，并且更好地了解这个动态系统是一个有趣的问题。我们在尺寸上获得最佳结果$d\ge 3$：使用 $G（X，ÿ）$ 为了表示在紧凑流形上拉普拉斯算子的格林函数，我们表明 $X_{ñ}=精氨酸\underset{X\in \mathrm{中号}}{分}\sum _{ķ=\mathrm{1个}}^{ñ-\mathrm{1个}}G（X，X_{ķ}）\text{满足}{\mathrm{w\; ^}}_2\left(\frac{\mathrm{1个}}{ñ}\sum _{ķ=\mathrm{1个}}^{ñ}{\delta }_{X_{ķ}}，dX\right)\lesssim \frac{\mathrm{1个}}{{ñ}^{\mathrm{1个}∕d}}。$