Problem 991

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claims/: The 2 claim pages of Problem 991, one per claimant's result; the problem's standing derives from them.

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Statement. Suppose $A=\{w_1,\ldots,w_n\}\subset S^2$ maximises

$$\prod_{i<j}\lvert w_i-w_j\rvert$$

over all possible sets of size $n$.

Is it true that

$$\max_C\lvert \lvert A\cap C\rvert - \alpha_C n\rvert =o(n),$$

where the maximum is taken over all spherical caps $C$ and $\alpha_C$ is the area of $C$ (normalised so that the entire sphere has area $1$)?

Status. Proved. The site labels the problem proved (page last edited 2025-09-16) on the strength of two refereed papers, while remarking that the attribution is unclear: Brauchart (Math. Comp. 2008) proves the quantitative bound $\max_C\lvert\lvert A\cap C\rvert-\alpha_C n\rvert\ll n^{3/4}$ and regards the equidistribution itself as classical potential theory; Marzo and Mas (Constr. Approx. 2021) prove $\ll n^{2/3}$ as the case $d=2$, $s=0$ of a bound for Riesz energy minimizers, a rate they credit to an unpublished manuscript of Wolff from around 1992. Each rate answers the question, so the standing is solved/proved, derived from the two accepted claim pages Brauchart 2008 and Marzo and Mas 2021; the acceptance evidence on each is the refereed publication and the site's acceptance, not a review by this corpus.

Source. erdosproblems.com/991, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #991, https://www.erdosproblems.com/991.

References.

• [Br08] Brauchart, J. S., Optimal logarithmic energy points on the unit sphere. Math. Comp. (2008), 1599-1613.
• [MaMa21] Marzo, Jordi and Mas, Albert, Discrepancy of minimal Riesz energy points. Constr. Approx. (2021), 473-506.

Formalization. None built or audited here. A public Lean 4 development in Boris Alexeev's lean-proofs collection declares itself a formalization of Marzo and Mas's solution and proves the qualitative $o(n)$ statement without a rate; it is linked, pinned, on their claim page. The site records no formal-conjectures statement file, and the community database lists the problem as unformalized. The OpenAI mathematics release of September 2026 states, in its preprints on the planar Coulomb renormalized energy and on the universal optimality of the triangular lattice (release at the pinned revision), results about point configurations in the plane, computer-assisted and not verified here, with Lean developments for the universal optimality and a planar packing certificate that this corpus has not built; they do not state this problem and touch its maximizers only through the asymptotics of the minimal logarithmic energy on $S^2$, not through their cap discrepancy, so no claim page records them.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• marzo_2021_discrepancy_minimal_riesz_energy_points
• marzo_2021_discrepancy_minimal_riesz_energy_points / theorem_1_1
• marzo_2021_discrepancy_minimal_riesz_energy_points / theorem_1_5
• openai_2026_atomic_certificate_triangular_lattice_universal_optimality
• openai_2026_atomic_certificate_triangular_lattice_universal_optimality / theorem_1_1
• openai_2026_atomic_certificate_triangular_lattice_universal_optimality / theorem_1_2
• openai_2026_sharp_fourier_certificate_planar_circle_packing
• openai_2026_sharp_fourier_certificate_planar_circle_packing / theorem_1_1
• openai_2026_triangular_minimality_planar_coulomb_renormalized_energy
• openai_2026_triangular_minimality_planar_coulomb_renormalized_energy / corollary_1_3
• openai_2026_triangular_minimality_planar_coulomb_renormalized_energy / theorem_1_1
• openai_2026_universal_optimality_triangular_lattice
• openai_2026_universal_optimality_triangular_lattice / corollary_8_1
• openai_2026_universal_optimality_triangular_lattice / theorem_1_1