Equilibrium point configurations on the line

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $(a_n)_{n\in\mathbb Z}$ be a strictly increasing, locally finite sequence of points on the real line, with $a_n\to\pm\infty$ as $n\to\pm\infty$. Each point represents an identical particle, and the repulsive force at distance $r>0$ has magnitude $r^{-2}$. The force from each side is required to be finite: \[\sum_{i<n}(a_n-a_i)^{-2}<\infty,\qquad \sum_{i>n}(a_i-a_n)^{-2}<\infty.\] Mechanical equilibrium at $a_n$ means equality of these two sums, or equivalently $\sum_{i\ne n}|a_i-a_n|^{-2}\operatorname{sgn}(a_i-a_n)=0$, where $\operatorname{sgn}$ is the sign function.

Statement recorded in the supplied source.

If every particle is in equilibrium, must there exist $\alpha>0$ and $\beta\in\mathbb R$ such that $a_n=\alpha n+\beta$ for every $n\in\mathbb Z$? The question includes aperiodic configurations and does not assume a global bound on consecutive gaps. [S2]

Short English statement

Must an infinite equilibrium configuration of identical particles with inverse-square repulsion be equally spaced?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 10000003 (AMR-099-0003). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Agelos Georgakopoulos and Mihail N. Kolountzakis, On particles in equilibrium on the real line, arXiv:1604.01649v1 (2016), Section 1, Problem 1 and definition of equilibrium, pp. 1--2. https://arxiv.org/pdf/1604.01649.