Equilibria of Inverse-Square Repulsion on the Line Are Arithmetic Progressions

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 $1<s\le 2$, 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 ${\sum }_{y\in X\setminus \{x\}}\mathrm{\mid }y-x{\mathrm{\mid }}^{-s}$ is finite and the net force ${\sum }_{y\in X\setminus \{x\}}\mathrm{sgn}(y-x)\mathrm{\mid }y-x{\mathrm{\mid }}^{-s}$ 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 $1.5386\dots$ when $s=2$. With this bound, an energy estimate shows that the Doob transform of the walk by $g$ is recurrent. Since $1\mathrm{/}g$ is a positive harmonic function of the transformed walk, it is constant.