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Statement

Setting (pp. 2, 4). Let $\sigma$ be the surface measure on $\mathbb{S}^d$ and $\omega_d=\sigma(\mathbb{S}^d)$. For $r\ge0$, $\mathbb{H}^r(\mathbb{S}^d)$ is the Sobolev space of $f\in L^2(\mathbb{S}^d)$ with $\sum_{\ell\ge0}\sum_{k=1}^{h_\ell}(1+\ell^2)^r\lvert f_{\ell,k}\rvert^2<\infty$, the $f_{\ell,k}$ being the coefficients of $f$ in an orthonormal basis of spherical harmonics, normed by the square root of that sum. For a Borel measure $\mu$ on $\mathbb{S}^d$, the dual norm $\lVert\mu\rVert_{\mathbb{H}^{-r}(\mathbb{S}^d)}$ is the supremum of $\int\psi\,d\mu$ over smooth $\psi$ with $\lVert\psi\rVert_{\mathbb{H}^r(\mathbb{S}^d)}=1$.

Definition 1.3 (pp. 4--5). For an $N$-point set $X_N=\{x_1,\ldots,x_N\}\subset\mathbb{S}^d$, $\epsilon>0$ and $0\le s<d$, put $D_j=D_{\epsilon N^{-1/d}}(x_j)$, the cap of centre $x_j$ and Euclidean radius $\epsilon N^{-1/d}$, and

$$\mu_{X_N,\epsilon}=\Bigl(\frac1N\sum_{j=1}^N\frac{\chi_{D_j}}{\sigma(D_j)}-\frac1{\omega_d}\Bigr)\sigma .$$

The Sobolev discrepancy of $X_N$ is $D^\epsilon_{s,d}(X_N)=\lVert\mu_{X_N,\epsilon}\rVert_{\mathbb{H}^{(s-d)/2}(\mathbb{S}^d)}$, the dual norm of order $(d-s)/2$. Remark 1.4 (p. 5) notes that Wolff used a homogeneous Sobolev norm instead and that the zero-order term is absorbed in the proof of Theorem 1.1.

Theorem 1.5 (p. 5). 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, for every $\epsilon>0$ small enough depending only on $d$ and $s$,

$$N^{-\frac12+\frac{s}{2d}}\lesssim D^\epsilon_{s,d}(X_N)\lesssim N^{-\frac1d}+N^{-\frac12+\frac{s}{2d}},$$

with implied constants depending only on $d$, $s$ and $\epsilon$. The paper calls the estimate sharp in the range $d-2\le s<d$ (p. 5), where the two sides have the same order.

Proof pointer

Section 4, pp. 21--24. Lemma 4.1 (p. 21) expands the energy of the smoothed measures $\mu_i=\chi_{D_i}\sigma/\sigma(D_i)$ for a rotation-invariant kernel; Proposition 4.2 (pp. 22--23) applies it to the Riesz kernel and evaluates the self-energy of a cap of radius $\epsilon N^{-1/d}$. Lemma 2.4 (p. 9) makes $E_s(h)$ comparable to $\lVert h\rVert^2_{\mathbb{H}^{(s-d)/2}(\mathbb{S}^d)}$. The lower bound (4.3) follows from these and the lower estimates for the minimal energy, (1.2) for $0<s<d$ and (1.3) for $s=0$; the upper bound adds Corollary 3.7 (p. 21), which compares the discrete minimal energy with the energy of the smoothed caps (p. 24).

Read depth

Claims checked: Definition 1.3, Theorem 1.5 and its proof on pp. 23--24 were read clause by clause on the page images of the print. The upper estimate on p. 24 is displayed for $0<s<d$ and for $s=0$ with $d>2$. For $d=2$ and $d-2\le s<d$, including $s=0$, Corollary 3.7 rests on Theorem 3.4 (p. 19, stated for $d>2$ and $0<s<d$) through Remark 3.6 (p. 20), which says the extension to $d\ge2$ and $0\le s<d$ is not hard and omits the details. Nothing here is independently reviewed.

Dependencies

None in the corpus. Inside the paper: Lemma 2.4, Corollary 3.7 (with Theorem 3.4, Remark 3.6 and Lemma 3.1), Lemma 4.1, Proposition 4.2, and the known asymptotics (1.2) and (1.3) of the minimal energy, which the paper cites rather than proves.

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: through Theorem 1.1, which the paper derives from this estimate; for $d=2$, $s=0$ the bound here reads $D^\epsilon_{0,2}(X_N)\lesssim N^{-1/2}$.