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Statement

Setting (pp. 1--2). For an $N$-point set $X_N=\{x_1,\ldots,x_N\}$ on the unit sphere $\mathbb{S}^d\subset\mathbb{R}^{d+1}$ and $0\le s<d$, the Riesz $s$-energy is $E_s(X_N)=\sum_{i\ne j}\lvert x_i-x_j\rvert^{-s}$ for $0<s<d$, and the logarithmic energy is $E_0(X_N)=\sum_{i\ne j}\log\frac{1}{\lvert x_i-x_j\rvert}$ for $s=0$. A minimizer is an $N$-point set attaining the infimum $\mathcal{E}_s(N)$ of $E_s$ over all $N$-point subsets of $\mathbb{S}^d$. The measure $\widetilde\sigma$ is the surface measure normalized to total mass $1$, and $D_r(x)=\{y\in\mathbb{S}^d:\lvert x-y\rvert<r\}$ is the spherical cap of centre $x$ and Euclidean radius $r>0$.

Theorem 1.1 (p. 3). Let $0\le s<d$ and let $X_N$ be an $N$-point set of minimizers of the Riesz $s$-energy on $\mathbb{S}^d$. Then

$$\sup_D\left\lvert\frac{\#(X_N\cap D)}{N}-\widetilde\sigma(D)\right\rvert \lesssim\chi_{[0,d-2]}(s)\,N^{-\frac{2}{d(d-s+1)}} +\chi_{(d-2,d)}(s)\,N^{-\frac{2(d-s)}{d(d-s+4)}},$$

the supremum taken over all spherical caps $D\subset\mathbb{S}^d$, with implied constants depending only on $d$ and $s$. Here $\chi_I$ is the indicator of the interval $I$: the first term is the bound for $0\le s\le d-2$, the second for $d-2<s<d$.

Remark 1.2 (p. 3). The paper states that the same bound holds when the discrepancy is taken over the $K$-regular sets of Sjögren (its reference [26]) instead of spherical caps; it gives no separate proof.

Context given by the paper (p. 3). The previously known bound for $s\ne d-1$ is Brauchart's $O(N^{-(d-s)/(d(d-s+2))})$ for $0\le s<d$, display (1.4). The paper says Theorem 1.1 improves it for $0\le s<2$ on $\mathbb{S}^2$, where $s=0$ is the $O(N^{-1/3})$ bound of an unpublished manuscript of Wolff, and for $d-t_0<s<d$ when $d\ge3$, with $t_0=\frac{1+\sqrt{17}}{2}$; the abstract excludes $s=1$ on $\mathbb{S}^2$ and $s=d-1$ for $d\ge3$, where Götz's $O(N^{-1/d}\log N)$ for the harmonic case $s=d-1$ remains the best bound. It notes that all these bounds are far from Beck's order $N^{-(d+1)/(2d)}$, up to a logarithmic term, for the optimal cap discrepancy of $N$-point sets on $\mathbb{S}^d$.

Proof pointer

Section 5, pp. 24--27. The paper proves Theorem 1.1 by combining Theorem 1.5, which bounds the Sobolev discrepancy $D^{\epsilon_0}_{s,d}(X_N)$ of a minimizer by a constant times $N^{-1/d}+N^{-1/2+s/(2d)}$, with Proposition 5.2 (pp. 25--27), which holds for every $N$-point set: a Sobolev discrepancy bound of that form, with constant $C_1$, implies the cap bound of Theorem 1.1 with a constant $C_2$ depending only on $d$, $s$, $\epsilon_0$ and $C_1$. Proposition 5.2 tests the measure $\mu_{X_N,\epsilon_0}$ against smooth functions $f^{\pm}_\epsilon$ squeezed between caps of radii differing by $O(\epsilon)$, controls the pairing with an interpolation inequality between Sobolev norms (Lemma 5.1, p. 24), and optimizes $\epsilon$, choosing $\epsilon=N^{-2(d-s)/(d(d-s+4))}$ for $d-2<s<d$ and $\epsilon=N^{-2/(d(d-s+1))}$ for $0\le s\le d-2$ (p. 27).

Read depth

Claims checked: the setting, Theorem 1.1, Remark 1.2 and the comparison with earlier bounds were read clause by clause on the page images of the print, and the deduction from Theorem 1.5 through Proposition 5.2 was followed for structure. Nothing here is independently reviewed. For the case $d=2$, $s=0$, see the read-depth note on Theorem 1.5.

Dependencies

Theorem 1.5 of the same paper, with Proposition 5.2 and Lemma 5.1.

Source. J. Marzo and A. Mas, Discrepancy of minimal Riesz energy points, Constr. Approx. 54 (2021), 473--506, doi:10.1007/s00365-021-09534-5; arXiv:1907.04814. Labels and page numbers here are those of arXiv:1907.04814v1, as named on the source card.

Bears on

• Problem 991: the $n$-point subsets of $S^2$ maximizing $\prod_{i<j}\lvert w_i-w_j\rvert$ are exactly the minimizers of the logarithmic energy $E_0$ on $\mathbb{S}^2$, the case $d=2$, $s=0$ of Theorem 1.1, where the first term applies and gives $\sup_D\lvert\#(A\cap D)/n-\widetilde\sigma(D)\rvert\lesssim n^{-1/3}$. Multiplying by $n$, the problem's quantity $\max_C\lvert\lvert A\cap C\rvert-\alpha_C n\rvert$ is $O(n^{2/3})$, so $o(n)$. The paper does not mention the problem; it credits this rate for $d=2$, $s=0$ to Wolff's unpublished manuscript.