贪婪能量最小化可以用二进制计数：点电荷和范德科普特序列
Greedy energy minimization can count in binary: point charges and the van der Corput sequence
摘要：
本文建立了一个问题之间的联系，在潜在的理论和数学物理，安排点，使最小化的能量泛函，和一个问题，在组合数学和数论，构建“均匀分布”序列的点$[0,1)$。令$f:[0,1] \rightarrow \mathbb{R}$是（i）对称的$f(x) = f(1-x)$，（ii）在$(0,1)$上二次可微，以及（iii）使得对于所有$x \in (0,1)$，$f''(x)>0$。我们研究了贪婪动力系统，其中，给定一个初始集$\{x_0, \ldots, x_{N-1}\} \subset [0,1)$，点$x_N$作为$$ x_{N} = \arg\min_{x \in [0,1)} \sum_{k=0}^{N-1}{f(|x-x_k|)}.$$获得。我们证明，如果我们开始这个建设与单元素$x_0=0$，那么所有出现的建设是置换的范德科普特序列（计数在二进制和反映的逗号）：\textit{greedy energy minimization recovers the way we count in binary.}这给出了一个新的建设经典的范德科普特序列。特殊情况$f(x) = 1-\log(2 \sin(πx))$回答了Steinerberger的一个问题。有趣的是，我们推导出的点集在另一个不同的上下文中也被称为单位圆盘上的Leja序列。此外，我们给出了一个一般的界限上的差异，以这种方式构造的函数$f$满足一个额外的假设。
英文摘要：
This paper establishes a connection between a problem in Potential Theory and Mathematical Physics, arranging points so as to minimize an energy functional, and a problem in Combinatorics and Number Theory, constructing 'well-distributed' sequences of points on $[0,1)$. Let $f:[0,1] \rightarrow \mathbb{R}$ be (i) symmetric $f(x) = f(1-x)$, (ii) twice differentiable on $(0,1)$, and (iii) such that $f''(x)>0$ for all $x \in (0,1)$. We study the greedy dynamical system, where, given an initial set $\{x_0, \ldots, x_{N-1}\} \subset [0,1)$, the point $x_N$ is obtained as $$ x_{N} = \arg\min_{x \in [0,1)} \sum_{k=0}^{N-1}{f(|x-x_k|)}.$$ We prove that if we start this construction with the single element $x_0=0$, then all arising constructions are permutations of the van der Corput sequence (counting in binary and reflected about the comma): \textit{greedy energy minimization recovers the way we count in binary.} This gives a new construction of the classical van der Corput sequence. The special case $f(x) = 1-\log(2 \sin(πx))$ answers a question of Steinerberger. Interestingly, the point sets we derive are also known in a different context as Leja sequences on the unit disk. Moreover, we give a general bound on the discrepancy of any sequence constructed in this way for functions $f$ satisfying an additional assumption.
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