发表机构
Mercury Software GmbH（Mercury软件有限公司）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明实数轴上平方反比排斥力作用下的平衡点构型必为等差数列，推广了先前结果，并利用随机游走调和函数与能量估计给出证明。
AI 中文摘要
Benjamini 曾问：在平方反比排斥力作用下处于平衡状态的实数轴上点构型是否必定为等差数列。Georgakopoulos 和 Kolountzakis 在相邻点之间的某个间隙具有最大或最小长度的情况下证明了该结论，并将一般（非周期）情形视为未解决问题。我们证明答案是肯定的。更一般地，设 $1<s\le2$，且 $X\subset\mathbb{R}$ 是一个局部有限集合，至少包含两个点，使得对每个 $x\in X$，总力 $\sum_{y\in X\setminus\{x\}}|y-x|^{-s}$ 有限，且净力 $\sum_{y\in X\setminus\{x\}}\operatorname{sgn}(y-x)\\,|y-x|^{-s}$ 为零。则 $X$ 是等差数列。不需要对间隙作任何先验假设。将两个相邻点的平衡方程相减，可看出间隙 $g_n$ 构成一个显式可逆随机游走在 $\mathbb{Z}$ 上（具有长程跳跃）的正调和函数。平衡还限制相邻间隙之比，当 $s=2$ 时上界为 $1.5386\ldots$。利用该界，一个能量估计表明该游走经 $g$ 的 Doob 变换是常返的。由于 $1/g$ 是变换后游走的正调和函数，故其为常数。
英文摘要
Benjamini asked whether every configuration of points on the real line that is in equilibrium under the inverse-square repulsive force must be an arithmetic progression. Georgakopoulos and Kolountzakis proved this when some gap between consecutive points has maximal or minimal length, and described the general (aperiodic) case as open. We show that the answer is yes. More generally, let $1<s\le2$, and let $X\subset\mathbb{R}$ be a locally finite set with at least two points such that, for every $x\in X$, the total force $\sum_{y\in X\setminus\{x\}}|y-x|^{-s}$ is finite and the net force $\sum_{y\in X\setminus\{x\}}\operatorname{sgn}(y-x)\,|y-x|^{-s}$ is zero. Then $X$ is an arithmetic progression. No a priori assumption on the gaps is needed. Subtracting the equilibrium equations of two consecutive points shows that the gaps $g_n$ form a positive harmonic function for an explicit reversible random walk on $\mathbb{Z}$ with long-range jumps. Equilibrium also bounds the ratio of consecutive gaps, by $1.5386\ldots$ when $s=2$. With this bound, an energy estimate shows that the Doob transform of the walk by $g$ is recurrent. Since $1/g$ is a positive harmonic function of the transformed walk, it is constant.
Comments10 pages. AI-assisted tools were used; see the paragraph "Use of AI tools". Earlier version: doi:10.5281/zenodo.23113437
