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Claim. Brauchart, Optimal logarithmic energy points on the unit sphere, Math. Comp. 77 (2008), no. 263, 1599–1613 (published electronically 2008-02-06), studies the $N$-point sets on $S^d\subset\mathbb{R}^{d+1}$, $d\ge2$, that maximize the product of all pairwise distances, equivalently minimize the logarithmic energy, states that they are uniformly distributed as $N\to\infty$, and quantifies this by bounding their spherical cap discrepancy

$$D_C(X_N^*)=\sup_C\left\lvert\frac{\lvert X_N^*\cap C\rvert}{N} -\sigma(C)\right\rvert = O\bigl(N^{-1/(d+2)}\bigr),$$

the supremum over all spherical caps $C$ and $\sigma$ the normalized surface measure. For $d=2$ this is $O(N^{-1/4})$, so the count of a maximizing $n$-point set in any cap differs from $\alpha_C n$ by $O(n^{3/4})$ uniformly over caps, which answers the question of Problem 991 in the affirmative with a rate. The paper's introduction (p. 1600) attributes the equidistribution itself to classical potential theory, citing Landkof's monograph: the logarithmic energy of probability measures on $S^d$ is uniquely minimized by $\sigma$.

Statements in the paper. Proposition 1, stated for $d\ge2$, says that optimal logarithmic energy $N$-point configurations are uniformly distributed as $N\to\infty$; the paper proves it in Subsection 2.1 by an argument it describes as not potential-theoretic and notes that it also follows from the discrepancy bound. Theorem 1.6, also stated for $d\ge2$, gives $D_C(X_N^*)=O(N^{-1/(d+2)})$, proved in Subsection 2.2 for more general, $K$-regular, test sets. The abstract's restriction to $d\ge3$ concerns only the paper's new second term $(1/d)(\log N)/N$ of the energy expansion, which was previously known for $S^2$ alone; the two distribution statements cover $S^2$, so the case the problem asks about is settled by the paper directly. Marzo and Mas's display (1.4), on the 2021 card, restates the bound as $O(N^{-(d-s)/(d(d-s+2))})$ for the Riesz $s$-energy minimizers on $S^d$, $0\le s<d$, citing this paper for the logarithmic case $s=0$, which is $O(N^{-1/4})$ on $S^2$. The paper is not held in the library.

Depends on. Nothing in this wiki; the result rests on the cited paper alone.

Acceptance. Refereed: Math. Comp. 77 (2008), 1599–1613, published electronically 2008-02-06. Reviewed: the site's curator, T. F. Bloom, lists the problem as proved and credits this paper with the rate $\ll n^{3/4}$ and with the remark that the qualitative statement is classical potential theory, while noting that the attribution is therefore unclear (site page last edited 2025-09-16). The thread's one comment (2025-10-17), by Terence Tao and not by the curator, reports a literature search with ChatGPT (the version the comment calls "Thinking") and the Gemini deep research tool that found no earlier published reference than this paper and describes the qualitative equidistribution as folklore among potential theorists, with Landkof's monograph among the standard texts the paper cites; it adds no acceptance evidence beyond the curator's label. This corpus has not reproduced the proof; the standing rests on the refereed paper and the site's acceptance.

Relation to the other claim. The rate here, $O(n^{3/4})$, is weaker than the $O(n^{2/3})$ of Marzo and Mas, which also credits the $O(n^{2/3})$ bound on $S^2$ to an unpublished manuscript of Wolff from around 1992. Either rate settles the $o(n)$ question; the two claims are independent proofs.