正定函数、指数和与贪婪算法：一个奇怪的现象
Positive-definite Functions, Exponential Sums and the Greedy Algorithm: a curious Phenomenon
摘要：
我们描述了一个奇怪的动力系统，结果在$[0,1]$中的实数序列似乎显着的性能。令函数$f:\mathbb{T} \rightarrow \mathbb{R}$满足$\hat{f}(k) \geq c|k|^{-2}$，并通过$$ x_n = \arg\min_x \sum_{k=1}^{n-1}{f(x-x_k)}.$$定义一个序列。这样的序列$(x_n)_{n=1}^{\infty}$似乎以各种方式被随机规则地分布（满足有利的指数和估计;每个区间$J \subset [0,1]$包含$\sim |J|n$元素）。我们证明了$$ W_2\left( \frac{1}{n} \sum_{k=1}^{n}{δ_{x_k}}, dx\right) \leq \frac{c}{\sqrt{n}},$$，其中$W_2$是2-Wasserstein距离。更强的结果似乎是真实的，这似乎是一个有趣的问题，以更好地理解这个动力系统。在维数$d \geq 3$上得到了最优结果：用$G(x,y)$表示紧致流形上拉普拉斯算子的格林函数，证明了$$ x_n = \arg\min_{x \in M} \sum_{k=1}^{n-1}{G(x,x_k)} \quad \mbox{satisfies} \quad W_2\left( \frac{1}{n} \sum_{k=1}^{n}{δ_{x_k}}, dx\right) \lesssim \frac{1}{n^{1/d}}.$$
英文摘要：
We describe a curious dynamical system that results in sequences of real numbers in $[0,1]$ with seemingly remarkable properties. Let the function $f:\mathbb{T} \rightarrow \mathbb{R}$ satisfy $\hat{f}(k) \geq c|k|^{-2}$ and define a sequence via $$ x_n = \arg\min_x \sum_{k=1}^{n-1}{f(x-x_k)}.$$ Such sequences $(x_n)_{n=1}^{\infty}$ seem to be astonishingly regularly distributed in various ways (satisfying favorable exponential sum estimates; every interval $J \subset [0,1]$ contains $\sim |J|n$ elements). We prove $$ W_2\left( \frac{1}{n} \sum_{k=1}^{n}{δ_{x_k}}, dx\right) \leq \frac{c}{\sqrt{n}},$$ where $W_2$ is the 2-Wasserstein distance. Much stronger results seem to be true and it seems like an interesting problem to understand this dynamical system better. We obtain optimal results in dimension $d \geq 3$: using $G(x,y)$ to denote the Green's function of the Laplacian on a compact manifold, we show that $$ x_n = \arg\min_{x \in M} \sum_{k=1}^{n-1}{G(x,x_k)} \quad \mbox{satisfies} \quad W_2\left( \frac{1}{n} \sum_{k=1}^{n}{δ_{x_k}}, dx\right) \lesssim \frac{1}{n^{1/d}}.$$
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