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 ≤ 2, and let X ⊂ ℝ be a locally finite set with at least two points such that, for every x ∈ X, the total force Σ_{y∈X∖{x}} |y − x|^(−s) is finite and the net force Σ_{y∈X∖{x}} sgn(y − x) |y − x|^(−s) is zero. Then X is an arithmetic progression. No 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 ℤ with long-range jumps. Equilibrium also bounds the ratio of consecutive gaps, by 1.5386… 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. This is an unrefereed note. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: AMR-099-0003 (I. Benjamini, "Points in equilibrium", note of May 2015, Question 0.1; also Question 5 in Section 4 of the 2015 Warwick open-problem list "Random walks on graphs and potential theory").