Polarization and Greedy Energy on the Sphere

TL;DR

This paper studies a greedy algorithm for placing points on a sphere to minimize Riesz energy, showing it achieves near-optimal energy behavior and providing bounds on discrepancy, with numerical evidence of high uniformity.

Contribution

It proves that greedy sequences on the sphere attain optimal second-order Riesz energy behavior for 0<s<d and relates polarization to energy and discrepancy.

Findings

Greedy sequences achieve optimal second-order Riesz energy for 0<s<d.

Second-order polarization term is of order N^{s/d} for 0<s<d.

Upper bounds on L^2 spherical cap discrepancy are established, with numerical evidence of low discrepancy.

Abstract

We investigate the behavior of a greedy sequence on the sphere $\mathbb{S}^d$ defined so that at each step the point that minimizes the Riesz $s$-energy is added to the existing set of points. We show that for $0<s<d$, the greedy sequence achieves optimal second-order behavior for the Riesz $s$-energy (up to constants). In order to obtain this result, we prove that the second-order term of the maximal polarization with Riesz $s$-kernels is of order $N^{s/d}$ in the same range $0<s<d$. Furthermore, using the Stolarsky principle relating the $L^2$-discrepancy of a point set with the pairwise sum of distances (Riesz energy with $s=-1$), we also obtain a simple upper bound on the $L^2$-spherical cap discrepancy of the greedy sequence and give numerical examples that indicate that the true discrepancy is much lower.

Equations

$$E_K({\omega }_{N,d})=2\sum _{1\le i<j\le N}K(x_i,x_j).$$

$$K_s(x,y):=\{\begin{array}{ll}\mathrm{sgn}(s)\Vert x-y{\Vert }^{-s}& s\ne 0\\ -\log (\Vert x-y\Vert )& s=0.\end{array}$$

$$P_K({\omega }_{N,d})=\underset{x\in {\mathbb{S}}^d}{\min }\sum _{j=1}^{N}K(x,x_j).$$

$${\mathcal{E}}_K(N)=\underset{{\omega }_{N,d}\subset {\mathbb{S}}^d}{\min }{E}_K({\omega }_{N,d})\text{and}{\mathcal{P}}_K(N)=\underset{{\omega }_{N,d}\subset {\mathbb{S}}^d}{\max }{P}_K({\omega }_{N,d}).$$

$$C(w,t)=\left\{x\in {\mathbb{S}}^d:⟨w,x⟩\ge t\right\}\subset {\mathbb{S}}^d$$

$$D({\omega }_{N,d})=\underset{w\in {\mathbb{S}}^d}{\sup }\underset{-1<t<1}{\sup }\left|\frac{1}{N}\sum _{i=1}^N{\chi }_{C(w,t)}(x_i)-{\sigma }_d(C(w,t))\right|,$$

$$N^{-\frac{1}{2}-\frac{1}{2d}}\lesssim \underset{{\omega }_{N,d}}{\inf }D({\omega }_{N,d})\lesssim N^{-\frac{1}{2}-\frac{1}{2d}}\sqrt{\log N},$$

$$D_{L^2,\text{mbox}cap}^2({\omega }_{N,d})={\int }_{-1}^1{\int }_{{\mathbb{S}}^d}{\left|\frac{1}{N}\sum _{i=1}^N{\chi }_{C(w,t)}(x_i)-{\sigma }_d(C(w,t))\right|}^{2}d{\sigma }_d(w)dt.$$

$$x_{N+1}=\operatorname*{\operatorname{argmin}}_{x\in\mathbb{S}^{d}}\sum_{i=1}^{n}K_{s}(x,x_{i})=\operatorname*{\operatorname{argmin}}_{x\in\mathbb{S}^{d}}E_{K_{s}}\big{(}\omega_{N,d}\cup\{x\}\big{)}.$$

$$\sum_{j=1}^{N}K_{s}(x_{N+1},x_{j})=P_{K_{s}}\big{(}\{x_{1},\dots,x_{N}\}\big{)}.$$

$${\mathcal{I}}_{s,d}={\int }_{{\mathbb{S}}^d}{\int }_{{\mathbb{S}}^d}{K}_s(x,y)d{\sigma }_d(x)d{\sigma }_d(y)={\int }_{{\mathbb{S}}^d}{K}_s(x,y)d{\sigma }_d(y),\forall x\in {\mathbb{S}}^d.$$

$$-b_{s,d}{N}^{\frac{s}{d}}\le {\mathcal{P}}_{K_s}(N)-{\mathcal{I}}_{s,d}N\le -c_{s,d}{N}^{\frac{s}{d}}.$$

$${\mathcal{I}}_{s,d}+b_{s,d}{N}^{\frac{s}{d}}\le {\mathcal{P}}_{K_s}(N)-{\mathcal{I}}_{s,d}N\le -c_{s,d}{N}^{\frac{s}{d}}.$$

$$-\frac{\log N}{d}-b_{0,d}\le {\mathcal{P}}_{K_0}(N)-{\mathcal{I}}_{0,d}N\le -c_{0,d}.$$

$${\mathcal{P}}_{K_s}(N)={\mathcal{I}}_{s,1}N+\frac{2\mathrm{sgn}(s)(2^s-1)}{(2\pi {)}^s}\zeta (s)N^s+\mathcal{O}(N^{s-2}).$$

$${\mathcal{P}}_{K_0}(N)=-\log (2).$$

$${\mathcal{I}}_{s,1}N-b_{s,1}{N}^s\le {\mathcal{P}}_{K_s}(N)\le {\mathcal{I}}_{s,1}N-c_{s,1}{N}^s.$$

$$-C_{s,d}{N}^{1+\frac{s}{d}}\le E_{K_s}({\omega }_{N,d})-{\mathcal{I}}_{s,d}{N}^2\le -C_{s,d}^{\prime }{N}^{1+\frac{s}{d}}.$$

$$C_{s,d}{N}^{1+\frac{s}{d}}\le E_{K_s}({\omega }_{N,d})-{\mathcal{I}}_{s,d}{N}^2\le -{\mathcal{I}}_{s,d}N-C_{s,d}^{\prime }{N}^{1+\frac{s}{d}}.$$

$$-\frac{1}{d}N\log (N)-C_{0,d}^{\prime }N\le E_{K_0}({\omega }_{N,d})-{\mathcal{I}}_{0,d}{N}^2\le \mathcal{O}(N).$$

$$E_{K_s}({\omega }_{N,d})\le {\mathcal{I}}_{s,d}(N^2-m^2)-C_{s,d}^{\prime }(N^{1+\frac{s}{d}}-m^{1+\frac{s}{d}})-{\mathcal{I}}_{s,d}(N-m)+E_{K_s}({\omega }_{m,d})+\mathcal{O}(N^{\frac{s}{d}}).$$

$$E_{K_s}({\omega }_{N,d})\le {\mathcal{I}}_{s,d}(N^2-m^2)-{\mathcal{I}}_{s,d}(N-m)-C_{s,d}^{\prime }(N^{1+\frac{s}{d}}-m^{1+\frac{s}{d}})+E_{K_s}({\omega }_{m,d})+\mathcal{O}(1).$$

$$E_{K_{-1}}({\omega }_{N,1})\le {\mathcal{I}}_{-1,1}(N^2-m^2)-{\mathcal{I}}_{-1,1}(N-m)-C_{-1,1}^{\prime }(\log (N)-\log (m))+E_{K_{-1}}({\omega }_{m,1})+\mathcal{O}(1).$$

$$E_{K_s}({\omega }_{N,1})\le {\mathcal{I}}_{s,1}(N^2-m^2)-{\mathcal{I}}_{s,1}(N-m)+E_{K_s}({\omega }_{m,1})+\mathcal{O}(N^{1+s})+\mathcal{O}(m^{1+s}).$$

$$\underset{N\to \infty }{\lim }\frac{E_{K_s}({\omega }_{N,d})}{N^2}={\mathcal{I}}_{s,d}.$$

$$\underset{N\to \infty }{\lim }\frac{P_{K_s}({\omega }_{N,d})}{N}={\mathcal{I}}_{s,d}.$$

$$\#\left\{1\le j\le N:P_{K_s}({\omega }_{j,d})\ge j\cdot {\mathcal{I}}_{s,d}-X_{s,d,\epsilon }\cdot j^{\frac{s}{d}}\right\}\ge \left(1-\epsilon \right)N\text{mbox}asN\to \infty .$$

$$\underset{x\in {\mathbb{S}}^d}{\max }\sum _{i=1}^n\Vert x-x_i\Vert \ge n{\int }_{{\mathbb{S}}^d}\sum _{i=1}^n\Vert x-x_i\Vert d{\sigma }_d(x)=n\cdot (-{\mathcal{I}}_{-1,d}),$$

$$\sum _{i=1}^n\Vert x_{n+1}-x_i\Vert \ge n\cdot (-{\mathcal{I}}_{-1,d}).$$

$$D_{L^2,\text{cap}}({\omega }_{N,d})\le \sqrt{\frac{1}{d}\frac{\Gamma ((d+1)/2)}{\sqrt{\pi }\Gamma (d/2)}}(-{\mathcal{I}}_{-1,d}{)}^{1/2}{\left(\frac{1}{N}+\frac{m^2-m}{N^2}\right)}^{1/2}.$$

Full text

1. Introduction and background

We consider the classical problem of distributing points on the sphere $\mathbb{S}^{d}$ as regularly as possible. Any such discussion requires a notion of regularity. We recall several such notions below.

1.1. Energy and polarization

Let $\omega_{N,d}=\left\{x_{1},\dots,x_{N}\right\}\subset\mathbb{S}^{d}$ denote a set of $N$ points in the sphere. For a symmetric, lower semi-continuous kernel $K:\mathbb{S}^{d}\times\mathbb{S}^{d}\rightarrow(-\infty,\infty]$ and a point set $\omega_{N,d}$, we define the discrete energy as

[TABLE]

Of particular importance are the Riesz $s$-kernels:

[TABLE]

Another quantity closely related to energy is polarization, defined as

[TABLE]

For a fixed $N$, we define the minimal energy and maximal (constrained) polarization, respectively, by

[TABLE]

Both energy (see e.g. [10, 11, 25, 49, 53]) and polarization (see e.g. [22, 42, 43, 70, 80, 79, 85]), especially in the case of Riesz kernels, have been actively studied over the past few decades: we refer the reader to the excellent recent book [20] for a comprehensive exposition. The second-order asymptotic behavior for the optimal Riesz $s$-energy is well understood (see the discussion in Section 4.2) for $-2<s<d$. However, results for second-order asymptotics of maximal polarization with Riesz $s$-kernels have not been obtained before (except when $d=1$). We prove these optimal bounds in Theorem 2.1 for the singular potential-theoretic range $0<s<d$ and then use this result to demonstrate that, in the same range, the greedy sequence obtains optimal second-order asymptotic behavior (up to constants) for Riesz $s$-energy (Theorem 2.2) and, for infinitely many $N$, polarization (Corollary 2.3).

In this paper we only consider the range $-2<s<d$. This is natural for the Riesz $s$-energy on the $d$-dimensional manifold $\mathbb{S}^{d}$, since for $s<-2$ minimizing energy no longer leads to uniform distribution (indeed, optimal configurations are achieved by placing half of the points at each of two opposite poles [16]), while for $s>d$ the kernels $K_{s}$ are hypersingular.

1.2. Spherical cap discrepancy

For any $w\in\mathbb{S}^{d}$ and any $t\in(-1,1)$, we define

[TABLE]

to be a spherical cap. The spherical cap discrepancy of a set $\omega_{N,d}=\left\{x_{1},\dots,x_{N}\right\}\subset\mathbb{S}^{d}$ is then defined as

[TABLE]

where $\sigma_{d}$ is the surface measure on $\mathbb{S}^{d}$ normalized so that $\sigma_{d}(\mathbb{S}^{d})=1$. A fundamental result of Beck [6, 7] states that

[TABLE]

where the implicit constants depend on the dimension $d$. The upper bound in (1.3) follows from a probabilistic construction (jittered sampling). There are many probabilistic [1, 3, 4, 8, 9, 17, 28, 45] and deterministic [1, 41, 44, 45, 47, 56, 66, 67, 69, 86] constructions that have been investigated. Most of these explicit point sets have only been proven to have discrepancy of the order no better than $N^{-1/2}$, although there is numerical evidence that some of them might be almost optimal. Instead of the spherical cap discrepancy, we will in this paper work with the $L^{2}-$spherical cap discrepancy

[TABLE]

The $L^{2}$-spherical cap discrepancy is smaller than the spherical cap discrepancy though the difference is, for many examples, at most logarithmic. The $L^{2}$-discrepancy is connected to energy, particularly to the Riesz energy with $s=-1$ (sum of distances) via the Stolarsky invariance principle [91], which we state in §5.

2. Greedy sequences and main results

One of the main purposes of this paper is to discuss a simple greedy construction of sequences that turns out to simultaneously

1. (i)

achieve optimal Riesz $s$-energy, up to the second-order behavior, when $0<s<d$, 2. (ii)

satisfy good distribution properties in the sense of $D_{L^{2},\tiny\mbox{cap}}^{2}$.

Greedy sequences have been studied in the context of energy [29, 52, 61, 62, 63, 64, 65, 73, 74, 82, 84, 95] and discrepancy [29, 57, 87, 88, 89, 90]. It is well understood that greedy sequences on $\mathbb{S}^{1}$ tend to be fairly structured [71], but this cannot be expected in higher dimensions. However, we show that greedy sequences on $\mathbb{S}^{d}$ exhibit an unusual amount of regularity: they produce nearly optimal Riesz energy for $0<s<d$ and their $L^{2}$-spherical cap discrepancy satisfies favorable bounds (while numerics in §6 suggests they might even be optimal).

Generally, constructing a sequence with good discrepancy or energy properties is significantly harder than constructing point sets for arbitrary $N$ since one is forced to keep previously chosen points. In some situations (e.g. discrepancy with respect to axis-parallel boxes on the torus [59, 72]) even the optimal discrepancy bounds are slightly different for sequences and point sets. Most known constructions of good point distributions on the sphere are not sequences. This makes the greedy sequence even more valuable.

2.1. Construction of the greedy sequence

We will work with an infinite sequence $(x_{n})_{n=1}^{\infty}$ instead of a single set. This is more difficult since points, once placed, remain fixed. We start with a completely arbitrary initial set of points $\left\{x_{1},\dots,x_{m}\right\}\subset\mathbb{S}^{d}$. The construction is then greedy: at each step we add the point that minimizes the Riesz $s$-energy with respect to the existing set of points. More formally, given a set $\omega_{N,d}=\left\{x_{1},\dots,x_{N}\right\}\subset\mathbb{S}^{d}$, we pick

[TABLE]

In other words, $x_{N+1}$ is just the point where the polarization is achieved for the set $\{x_{1},\dots,x_{N}\}$, i.e.

[TABLE]

If the minimum is not assumed at a unique point, then any such point may be chosen. Note that if $s=-1$, this construction is equivalent to maximizing the sum of (Euclidean) distances to the points in the existing set. When $s=0$, i.e. $K_{0}(x,y)=-\log(\|x-y\|)$, and $d=1$ (or more generally, on any compact $\Omega\subset\mathbb{C}$), such sequences are called Leja sequences in recognition of the work of Leja [60] although they were introduced by Edrei [39] earlier. Simultaneously, Górski studied the case of $s=-1$ (the Newtonian kernel) for compact sets $\Omega\subset\mathbb{R}^{3}$ [46]. For $s=d-2$ (i.e. the Green kernel) and compact $\Omega\subset\mathbb{R}^{d}$, the corresponding greedy sequences are known as Leja–Górski sequences, which have been studied in [48, 74]. Leja points have applications in Stochastic Analysis [68] and Approximation/Interpolation Theory [12, 30, 31, 33, 38, 54, 78, 92], and various numerical methods have been developed to approximate such points (see, e.g. [5, 21, 34]).

As a general remark on the behavior of greedy sequences in any dimension, we note that greedy sequences defined above (when initialized with a single point) have the property that every other point in the sequence is the antipodal point to the last one placed. This is a generalization of [63, Theorem 2.1], the proof is identical.

Proposition 2.1**.**

Let $s\in\mathbb{R}$. Let $(x_{n})_{n=1}^{\infty}$ be a greedy sequence on $\mathbb{S}^{d}$ for some initial point $x_{1}\in\mathbb{S}^{d}$. Then for all $k\in\mathbb{N}$, $x_{2k}=-x_{2k-1}$.

More generally, this result holds if we replace $K_{s}$ with any kernel $K(x,y)=f(\|x-y\|)$ with $f$ strictly decreasing.

2.2. Optimal second-order polarization estimates

For $s<d$, we note that the kernel $K_{s}(x,y)$ is integrable on $\mathbb{S}^{d}$ in each variable and depends only on the distance between $x$ and $y$. This suggests defining the constant

[TABLE]

with $\sigma_{d}$ denoting, as before, the normalized uniform measure on $\mathbb{S}^{d}$. Observe that the constant $\mathcal{I}_{s,d}$ is negative for $s<0$.

One would expect that polarization is maximized when the points are distributed as uniformly as possible. In that case, one could replace summation by integration in (1.2) and expect the maximal Riesz polarization $\mathcal{P}_{K_{s}}$ to behave roughly as $\mathcal{I}_{s,d}N$, and this is indeed correct, see e.g. [64]. We prove more delicate polarization estimates with optimal (up to constants) second-order terms for $0<s<d$.

Theorem 2.1**.**

For $-2<s<d$, there exist positive constants $b_{s,d},c_{s,d}$ such that the following hold for all $N\in\mathbb{N}$.

For $0<s<d$,

[TABLE]

For $-2<s<0$,

[TABLE]

For $s=0$,

[TABLE]

The lower bounds in the Theorem above (proved in §4.2) follow from known optimal energy estimates (see Theorem 4.1), but the upper bounds are novel and rely on the estimates of the generalized $L^{2}$ discrepancy [13, 14, 15], see §4.1.

Note that even in the previously studied case of the circle ($d=1$), our result is new when $-2<s<-1$. This in fact leads us to the following characterization of asymptotics for polarization on the circle:

Corollary 2.1**.**

For $-1\leq s<1$, $s\neq 0$,

[TABLE]

For $s=0$,

[TABLE]

When $-2<s<-1$, there are $b_{s,1},c_{s,1}\in\mathbb{R}_{>0}$ so that for $N$ sufficiently large,

[TABLE]

An extensive discussion of the one-dimensional case is presented in §3.

As suggested by these sharp bounds in the one-dimensional case, as well as the asymptotics for the hypersingular case $s\geq d$, which were studied in [2, 18, 19, 40, 51, 50], we conjecture that the upper bounds for polarization given in Theorem 2.1 are sharp for $-2<s\leq 0$ in all dimensions $d\geq 1$, i.e. the second term is of the order $\mathcal{O}(N^{s/d})$.

2.3. Riesz energy of greedy sequences

While Theorem 2.1 is interesting in its own right, we also use it as a tool to obtain second-order energy estimates for the greedy sequences (see §4.3), which are of optimal order in the range $0<s<d$.

Theorem 2.2**.**

Let $d\geq 2$, $-2<s<d$, and let $(x_{n})_{n\in\mathbb{N}}$ be a greedy sequence for $K_{s}$ as defined in (2.1) with fixed initial point $x_{1}\in\mathbb{S}^{d}$. For $N>1$, denote the first $N$ elements of the sequence by $\omega_{N,d}=\{x_{1},\dots,x_{N}\}$. Then there exist positive constants $C_{s,d},C^{\prime}_{s,d}$ such that the following holds for $N\geq 2$.

For $0<s<d$,

[TABLE]

For $-2<s<0$,

[TABLE]

For $s=0$,

[TABLE]

The results of Theorem 2.2 are new for all $d\geq 2$. The case of $s\leq-2$ was entirely handled in [63, Theorems 4.1 and 5.2] and the case of $s\geq d$ has been studied in [64, 65]. In the case of the circle $\mathbb{S}^{1}$, i.e. $d=1$, more precise information about the Riesz energy of greedy sequences is known, see Theorem 3.2.

The lower bounds in Theorem 2.2 are well-known optimal estimates for minimal Riesz $s$-energies discussed in §4.2, while the upper bounds for greedy sequences are the case $m=1$ of the following more general result about greedy sequences with an arbitrary number of initial points, which we prove in §4.3.

Theorem 2.3**.**

Let $d\in\mathbb{N}$, $-2<s<d$, and let $(x_{n})_{n\in\mathbb{N}}$ be the greedy sequence of points on $\mathbb{S}^{d}$ with respect to $K_{s}$ as defined in (2.1) with fixed arbitrary $x_{1},...,x_{m}\in\mathbb{S}^{d}$. Let $\omega_{N,d}$ denote the set of the first $N$ points in this sequence. Then there exists a positive constant $C_{s,d}$ such that for $N>m$, the following hold.

For $0\leq s<d$,

[TABLE]

For $-2<s\leq 0$ if $d\geq 2$ and $-1<s\leq 0$ if $d=1$,

[TABLE]

For $d=1$ and $s=-1$,

[TABLE]

For $d=1$ and $-2<s<-1$,

[TABLE]

These are the first second-order upper bounds for the energy in the literature for the case when the greedy sequence is initialized with $m$ arbitrary points, rather than a single point. The Riesz energy of greedy sequences initialized by one point has been studied before in a general setting, see e.g. [61, 64, 84], where the first-order term was obtained, along with the second-order estimates in the case $d=1$, see [63, 65]. In fact, for $m=1$ these results give much stronger bounds on the Riesz energy of greedy sequences on the circle $\mathbb{S}^{1}$, but the proofs rely strongly on the structural properties of the greedy sequences (see Theorem 3.2 and the discussion in §3.2), which do not generalize to $m>1$. Hence, the general results of Theorem 2.3 are interesting (although probably not sharp for $s\in(-2,0]$) even in the one-dimensional case).

2.4. Polarization of greedy sequences

Theorem 2.3 demonstrates, in particular, that for any greedy sequence with $m$ initial data points

[TABLE]

Combining this with Theorem 4.2.2 and Corollaries 14.6.5 and 14.6.7 from [20], we have that any such greedy sequence is uniformly distributed and therefore yields optimal first-order asymptotics for polarization (which was shown for $m=1$ in [64, Thm 2.1] and [84, Lemma 3.1]).

Corollary 2.2**.**

Let $d\in\mathbb{N}$, $-2<s<d$, and $(x_{n})_{n\in\mathbb{N}}$ be the greedy sequence of points on $\mathbb{S}^{d}$ with respect to $K_{s}$ as defined in (2.1) with fixed arbitrary $x_{1},...,x_{m}\in\mathbb{S}^{d}$. Let $\omega_{N,d}$ denote the set of the first $N$ points in this sequence. Then the sequence of point sets $\{\omega_{N,d}\}_{N=1}^{\infty}$ is uniformly distributed on $\mathbb{S}^{d}$, and

[TABLE]

Moreover, Theorem 2.2 states that greedy sequences on $\mathbb{S}^{d}$ achieve Riesz energy that is asymptotically optimal to the second-order (at least up to constants) in the potential-theoretic case $0<s<d$. This may be interpreted as a way of measuring regularity of a set of points, we would expect the greedy sequence to behave fairly regularly also with respect to other notions of uniformity. We prove that greedy points also achieve almost maximal polarization (up to the second-order term) most of the time in the sense of asymptotic density (in particular, for infinitely many $N$).

Corollary 2.3**.**

Let $d\geq 2$, $0<s<d$, and $(x_{n})_{n\in\mathbb{N}}$ be a greedy sequence for $K_{s}$. Let $\omega_{N,d}$ denote the set of the first $N$ points in this sequence. For every $\varepsilon>0$, there exists $X_{s,d,\varepsilon}>0$ such that

[TABLE]

This is optimal up to the value of $X_{s,d,\varepsilon}$ since Theorem 2.1 implies the unconditional bound $P_{K_{s}}(\omega_{j,d})\leq\mathcal{P}_{K_{s}}(j)\leq\mathcal{I}_{s,d}\cdot j-c_{s,d}\cdot j^{\frac{s}{d}}.$ The proof gives a quantitative description of how $X_{s,d,\varepsilon}$ depends on $s,d$ and $\varepsilon$ in terms of the constant $C_{s,d}$ arising in (2.9), we refer to the proof in §4.4 for details.

2.5. A uniform $L^{2}$-discrepancy bound

We now concentrate on Riesz energy with $s=-1$ or, equivalently, on the problem of selecting $\left\{x_{1},\dots,x_{N}\right\}\subset\mathbb{S}^{d}$ in such a way that the sum of distances $\sum_{i,j=1}^{N}\|x_{i}-x_{j}\|$ is maximized. The problem has particular geometric significance in light of the Stolarsky invariance principle (formally stated in §5) which states that maximizing the sum of distances is the same as minimizing the spherical $L^{2}-$cap discrepancy. On $\mathbb{S}^{2}$, there are a large number of deterministic constructions [1, 41, 44, 45, 47, 56, 66, 67, 69, 86] and some of them are known to achieve a spherical cap discrepancy as small as $N^{-1/2}$. The seminal results of Beck imply that optimal constructions should be as small as $N^{-3/4}$ (and there is numerical evidence that some of these sequences of point sets achieve it). Our contribution to this question is two-fold:

1. (1)

we show that a large class of recursively defined sequences, containing all greedy sequences, achieve an $L^{2}-$spherical cap discrepancy of order $N^{-1/2}$ with an explicit small constant, and 2. (2)

we provide some numerical evidence that greedy sequences achieve a rate that is either $N^{-3/4}$ or very close to it, see §6.

The result will apply to a broader class of sequences: we note the trivial inequality

[TABLE]

where we note that $-\mathcal{I}_{-1,d}>0$. The subsequent theorem applies to all sequences where the next element $x_{n+1}$ is always chosen in such a way that the inequality above is satisfied, i.e. rather than maximizing the sum, we choose any point where the sum exceeds its mean value (and, in particular, the greedy constructions always satisfy the inequality).

Theorem 2.4**.**

Let $\omega_{N}=\left\{x_{1},\dots,x_{m}\right\}\subset\mathbb{S}^{d}$ be an arbitrary initial set and suppose that for all $n\geq m$, the set is extended to a sequence satisfying

[TABLE]

Let $\omega_{N,d}$ denote the set of the first $N$ points in this sequence. Then, for any $N\geq m$,

[TABLE]

On $\mathbb{S}^{2}$, this expression simplifies to

[TABLE]

Observe that, due to the Stolarsky principle (Theorem 5.1), when $m=1$, the right-hand side this inequality exactly matches the first-order term in the upper bound in (2.10) in Theorem 2.2. In fact, inequality (2.10) is even stronger, since it includes a negative second-order term (similar conclusions for all $m\geq 1$ follow from (2.13)). However, this theorem still provides new information since the bounds here apply to a much wider class of sequences than just purely greedy sequences.

We first note that it applies to a fairly large family of sequences and gives a uniform $N^{-1/2}-$bound for all of them. This $N^{-1/2}-$bound is generically tight. One might be inclined to believe that the greedy sequence is better behaved, which is suggested by the case $0<s<d$ of Theorem 2.2.

Theorem 2.1, in the case $s=-1$, can be rewritten as saying that there is some $c_{d}>0$ such that for any $N$-point set $\{x_{1},\cdots,x_{N}\},$

[TABLE]

While this is sharp for $d=1$ and we believe it to be sharp for higher dimensions, it cannot be sharp for a greedy sequence “on average”, which becomes evident from the following result proved in §5.1 (recall that $-\mathcal{I}_{-1,d}=\frac{4}{3}$).

Proposition 2.2**.**

Let $(x_{n})$ be a greedy sequence on $\mathbb{S}^{2}$ with $m\geq 1$ initial points. For $N$ sufficiently large, depending on $m$,

[TABLE]

If greedy sequences indeed have good $L^{2}-$discrepancy properties (as suggested by numerics) and if $D_{L^{2},\emph{cap}}(\omega_{N,2})\ll N^{-1/2}$, then the upper bound in Proposition 2.2 has to be asymptotically tight. The gap between $1/25$ and $2/3$ is a quantitative way of measuring our lack of understanding of the underlying dynamics.

The proof of Proposition 2.2 can be easily adapted, with different constants, to other $s$-Riesz kernels and to all dimensions $d\geq 2$ (as well as for a wide range of more general kernels). We choose to state it just for $s=-1$ and $d=2$ for the sake of simplicity of exposition.

Remark 2.5*.*

The $L^{2}$ spherical cap discrepancy can also be interpreted as the worst-case error for numeric integration (with equal weights) for the Sobolev space $H^{\frac{d+1}{2}}(\mathbb{S}^{d})$, see [24], and our results can be translated to this setting. Greedy sequences in the context of worst-case error for numeric integration for reproducing kernel Hilbert spaces have been studied, for example, in [83], where a similar bound (although with an additional optimization over weights) was obtained.

2.6. Outline

The outline of the paper is as follows. In §3, we collect a number of results about both polarization and greedy sequences in the special case of the circle $\mathbb{S}^{1}$, presenting the proof of Corollary 2.1 as well as the analogues of Theorem 2.2 and Corollary 2.3 for $d=1$. In §4 we present the proofs of some of the main results for $d\geq 2$: Theorem 2.1 in §4.1–4.2, Theorems 2.2 and 2.3 in §4.3, and Corollary 2.3 in §4.4. In §5 we prove the $L^{2}$-discrepancy bound (Theorem 2.4) and Proposition 2.2 and discuss some similar known results. We conclude with discussion of numerical properties of greedy sequences in §6, and provide some auxiliary results in §7.

3. Polarization and Greedy Sequences on $\mathbb{S}^{1}$

The case of $d=1$ in Theorem 2.1 and Corollary 2.2 has mostly been settled in the literature previously, as the unique properties of the circle yield convenient examples of point sets that maximize polarization and greedy energy. In this section we collect known results on the circle for polarization and greedy energy and note where our Theorem 2.1 and Corollary 2.2 fill in some gaps or differ.

3.1. Polarization on the circle and proof of Corollary 2.1

In [51, Theorem 1] it is shown that equally spaced points on $\mathbb{S}^{1}$, denoted $\omega_{N}^{*}$, are optimal for polarization for any kernel $K(x,y)=f(\arccos(\langle x,y\rangle))$ such that $f$ is decreasing and convex on $[0,\pi]$. This includes the Riesz kernels for $-1\leq s<\infty$, but not $s<-1$. However, for $-2<s<-1$, $\omega_{N}^{*}$ still has optimal asymptotic behavior, and likely maximizes polarization as well.

For the range $-2<s$, the points that minimize the discrete potential with respect to $\omega_{N}^{*}$ are the midpoints of an arch between two $N$th roots of unity. Because the midpoints of an arch between two $N$th roots of unity are themselves $2N$th roots of unity, there is a natural expression for the polarization on $\mathbb{S}^{1}$ in terms of the corresponding energies for $N$ and $2N$ equally spaced points:

[TABLE]

There is an explicit formula for the energy of $N$ equally spaced points on $\mathbb{S}^{1}$.

Theorem 3.1** (Theorem 1.1 in [26]).**

Let $s\in\mathbb{R}$, $s\neq 0,1,3,5,...,$ and let $q$ be any nonnegative integer such that $q\geq(s-1)/2$. If $\omega_{N}^{\star}$ is a configuration of $N$ equally spaced points on $\mathbb{S}^{1}$, then

[TABLE]

where $\zeta(s)$ is the classical Riemann zeta function and $a_{n}(s)$ are the coefficients in the expansion

[TABLE]

Combining Theorem 3.1 with (3.1), we have that on $\mathbb{S}^{1}$ for $-2<s<1$, $s\neq 0$,

[TABLE]

This second order term also explicitly computed in [62, Proposition 4.1] and [63, Lemma 3.10]. In [26, page 623], the authors also show that

[TABLE]

which, when combined with (3.1), gives us that

[TABLE]

Both (3.2) and (3.4) yield a lower bound on optimal polarization $\mathcal{P}_{K_{s}}(N)\geq P_{K_{s}}(\omega_{N}^{*})$, which matches the upper bounds given in Theorem 2.1 for $-2<s<1$. In the case $-1\leq s<1$, optimality of roots of unity, [51, Theorem 1] shows that $\mathcal{P}_{K_{s}}(N)=P_{K_{s}}(\omega_{N}^{*})$, completing the proof of (2.6) and (2.7).

The proof of the upper bounds in Theorem 2.1 (which is presented in §4.1 and holds for all $d\geq 1$) covers the range $-2<s<-1$ in which the roots of unity are not known to be optimal, proving (2.8). Thus this case of Theorem 2.1 is new even in the one-dimensional case. This completes the proof of polarization estimates for $d=1$, i.e. Corollary 2.1.

3.2. Energy for greedy sequences

The behavior of greedy sequences on $\mathbb{S}^{1}$ is also well-studied. For $-2<s$, it is known that any greedy sequence is in fact a classical van der Corput sequence [35] (see [12, Theorem 5], [63, Lemma 3.7], [65, Sec 1.2], [64, Lemmas 4.1 and 4.2], and also [95, Example 2], which is perhaps the earliest observation of this kind, but just for $s=-1$). A similar result was shown for a large class of kernels (whenever $K(x,y)=f(\arccos(\langle x,y\rangle))$, and $f$ is a bounded, continuous, decreasing, convex function of the geodesic distance) in [71, Thm 2.1]. This has made explicit computations for bounds of the asymptotic behavior of the greedy Riesz energies possible (see Theorems 1.1, 1.2, and 1.5 in [65] and Theorems 3.16, 3.17, 3.18 in [63]). Here we collect the these results, in less detail.

Theorem 3.2**.**

On the circle $\mathbb{S}^{1}$, for $-2<s<1$ and $N\geq 2$, if $\omega_{N}$ is the first $N$ elements of a greedy sequence, then

[TABLE]

The order (of the second-order term) in each case cannot be improved.

We note that for $-2<s\leq 0$, this is an improvement on the upper bounds achieved in Theorem 2.2, suggesting that there is likely room for improvement on the upper bounds in higher dimensions in this range.

Moreover, according to Theorem 4.1, which states that the optimal second-order term for the Riesz $s$-energy on $\mathbb{S}^{d}$ is $\mathcal{O}(N^{1+\frac{s}{d}})$, we see that for $s>-1$, point sets produced via the greedy algorithm have optimal asymptotic behavior on $\mathbb{S}^{1}$, whereas for $-2<s\leq-1$, the greedy algorithm does not produce point sets with optimal Riesz energy, which leads us to the open question of whether these results also hold true for higher dimensions.

At the same time, at least when $s=-1$, these bounds are actually sharp for sequences. Indeed, the case of $s=-1$ in Theorem 3.2 and the Stolarsky invariance principle (Theorem 5.1) imply that the $L^{2}$-spherical caps discrepancy satisfies $D_{L^{2},cap}(\omega_{N})=\mathcal{O}(\sqrt{N^{-1}\log N})$. But spherical caps on the circle are just intervals, hence, this is just the classical periodic $L^{2}$-discrepancy of one-dimensional sequences, for which $\mathcal{O}(N^{-1}\sqrt{\log N})$ is known to be the optimal order, as shown in [75] (see also [55, 72]) with the ideas going back to the seminal results of Roth [81]. In fact, as mentioned earlier, in this case, the greedy sequence with one initial point is just the van der Corput sequence, whose discrepancy is well studied [32, 57, 76].

On the other hand, Theorem 4.1 shows that the optimal second-order term for the energy in the case $d=1$ and $s=-1$ is $\mathcal{O}(N^{1+\frac{s}{d}})=\mathcal{O}(1)$, and therefore, the optimal $L^{2}$-discrepancy is $\mathcal{O}(N^{-1})$ (by the Stolarsky principle), which is easily seen to be achieved by $N$ equally spaced points on the circle. This highlights an important difference between the behavior of $N$-point sets and infinite sequences.

3.3. Polarization for greedy sequences

As discussed earlier, the fact that any greedy sequence is uniformly distributed suggests that they may behave well for different measures of uniformity. For polarization, the asymptotic behavior of a greedy sequence for Riesz kernels on the circle was shown in [63, Theorem 3.11] and [62, Theorems 1.1 and 1.4] (the case of hypersingular Riesz kernels was also handled in [62] and the case of $s\leq-2$ in [63, Theorem 4.1 and 5.2]).

Theorem 3.3**.**

On the circle $\mathbb{S}^{1}$, for $-2<s<1$ and $N\geq 1$, if $\omega_{N}$ is the first $N$ elements of a greedy sequence, then

[TABLE]

The order (of the second-order term) in each case cannot be improved.

Comparing this to Corollary 2.1, we see that the greedy sequences on the circle have good second-order asymptotic behavior for $0<s$, but not for $-2<s\leq 0$.

4. The Proof of Theorems 2.1 and 2.2

We proceed with the proof of Theorem 2.1 for $d\geq 2$. We associate the Riesz $s$-kernel $K_{s}(x,y)$ with a function of the inner product $t$ between the points $x$ and $y$ by observing that $\|x-y\|=(2-2t)^{1/2}$ and setting $K_{s}(x,y)=f_{s}(\langle x,y\rangle)$, where:

[TABLE]

Further, to facilitate our proof of the upper bound for polarization, we also introduce non-singular approximations to the functions $f_{s}$. For $\varepsilon>0$, take

[TABLE]

4.1. Proof of Theorem 2.1: upper bound

For $d\in\mathbb{N}$, let $w_{d}(t)=(1-t^{2})^{\frac{d-2}{2}}$. For any $f\in L_{w_{d}}^{1}([-1,1])$, we have the Gegenbauer expansion

[TABLE]

where $C_{n}^{\lambda}$ are the standard Gegenbauer polynomials (see e.g. [36] for details) and

[TABLE]

Note that for all $s\in\mathbb{R}$ and $\varepsilon>0$, the function $f_{s,\varepsilon}$ is continuous, and therefore in $L_{w_{d}}^{2}([-1,1])$. On the other hand, $f_{s}\in L_{w_{d}}^{2}([-1,1])$ only when $s<\frac{d}{2}$, which would lead to certain technical complications in the proofs when $s\geq\frac{d}{2}$. At the same time $f_{s}\in L_{w_{d}}^{1}([-1,1])$ for all $s<d$, in other words, $K_{s}$ is integrable on the sphere, i.e. $\mathcal{I}_{s,d}<\infty$, for $s<d$ (this is the potential-theoretic case).

We shall need the information about the behavior of the Gegenbauer coefficients of the Riesz kernels $f_{s}$ as well as their approximations $f_{s,\varepsilon}$. We postpone a detailed analysis to the Appendix and only mention the most relevant information here: for $0<s<d$, the coefficients $\widehat{f}_{s,\varepsilon}(n,d)$ and $\widehat{f}_{s}(n,d)$ are non-negative and decreasing, and $\widehat{f}_{s}(n,d)$ is of the order $n^{s-d}$ (Corollary 7.2).

We start with the case $-2<s<\frac{d}{2}$ and will then explain the adjustments needed in the range $\frac{d}{2}\leq s<d$. Starting with an arbitrary $\omega_{N,d}=\{x_{1},\dots,x_{N}\}$ and trivially estimating the integral by the average, we observe that

[TABLE]

where we have used that due to rotational invariance the integral $\int_{\mathbb{S}^{d}}K_{s}(x,z)\,d\sigma_{d}(x)=\mathcal{I}_{s,d}$ is independent of $z\in\mathbb{S}^{d}$. Therefore,

[TABLE]

and we can thus estimate this expression from below by the $L^{2}$-average:

[TABLE]

where $D_{L^{2},f}(\omega_{N,d})$ is the generalized $L^{2}$-discrepancy of $\omega_{N,d}$ with respect to the function $f$ which is studied in [13, 14, 15] and well-defined for $f\in L^{2}_{w_{d}}([-1,1])$. Observe the similarity of this notion to the classical $L^{2}$-spherical cap discrepancy (1.4). Indeed, taking $f={\bf{1}}_{[t,1]}$, one obtains the inner integral in the definition (1.4) of $D_{L^{2},\tiny\mbox{cap}}(\omega_{N,d})$. It has been shown [13, Theorem 4.2] that

[TABLE]

for some constants $C_{d}$, $c_{d}>0$. Since $f_{s}\in L^{2}_{w_{d}}([-1,1])$ for $-2<s<d/2$ and Gegenbauer coefficients $\widehat{f_{s}}(n,d)$, $n>0$, are positive and decreasing (Corollary 7.2), we find that

[TABLE]

where we have used the asymptotics of the Gegenbauer coefficients $\widehat{f_{s}}(n,d)$. Taking the supremum over $\omega_{N,d}$ proves the upper bound in Theorem 2.1 for $-2<s<d/2$.

Turning to the case $s\geq d/2$, we can repeat the argument above verbatim for the kernel $K_{s,\varepsilon}(x,y)=f_{s,\varepsilon}(\langle x,y\rangle)$, using the fact that the Gegenbauer coefficients of $f_{s,\varepsilon}$ are positive and decreasing (Corollary 7.1) and the fact that $0\leq f_{s,\varepsilon}(t)\leq f_{s}(t)$), to find that

[TABLE]

Taking the limit as $\varepsilon\rightarrow 0$ which is justified by Lemma 7.3 and by the Lebesgue dominated convergence theorem, since $0\leq f_{s,\varepsilon}(t)\leq f_{s}(t)$. Using the asymptotics of $\widehat{f_{s}}(n,d)$ from Corollary 7.2, one obtains

[TABLE]

which proves the required bound for the range $d/2\leq s<d$.

4.2. Proof of Theorem 2.1: lower bound

We collect some known results from which the lower bound of Theorem 2.1 follows immediately for $d\geq 2$. The first relates the maximal polarization to the minimal energy.

Proposition 4.1** (Proposition 14.1.1, [20]).**

For all $N\geq 2$ and kernels $K$, we have

[TABLE]

Optimal second-order asymptotics for the discrete Riesz $s$-energy in the range $-2<s<d$ have been computed by various authors [23, 27, 58, 77, 93, 94], see also Theorems 6.4.5, 6.4.6, and 6.4.7 in [20]. These bounds are as follows.

Theorem 4.1**.**

For $d\geq 1$, $-2<s<d$, $s\neq 0$, there exist positive constants $C_{s,d}$, $C^{\prime}_{s,d}$ such that for $N\geq 2$,

[TABLE]

If $s=0$, then

[TABLE]

Combining Theorem 4.1 and Proposition 4.1, we deduce the following bounds.

Corollary 4.1**.**

For $d\in\mathbb{N}$, $-2<s<d$, $s\neq 0$, and $N\in\mathbb{N}$,

[TABLE]

where $C_{s,d}^{\prime}$ is as in Theorem 4.1.

If $s=0$, then

[TABLE]

Since the constant term is essential when $s<0$, this gives the lower bound of Theorem 2.1 for the case $d\geq 2$.

4.3. Greedy energy: proof of Theorems 2.2 and 2.3

The lower bounds in Theorem 2.2 are just the general lower bounds for minimal Riesz energies presented in Theorem 4.1 above, so we concentrate on the upper bound of Theorem 2.3.

Proof of Theorem 2.3.

Let $\omega_{N,d}$ denote the set of the first $N$ points of the greedy sequence with respect to $K_{s}$ with $m$ initial points. Observe that, by construction, for $k\geq m$

[TABLE]

Therefore the discrete energy of $\omega_{N+1,d}$ satisfies

[TABLE]

Using the upper bounds from Theorem 2.1 one obtains

[TABLE]

We have that for $M\geq 2$

[TABLE]

and since $N\geq m$, our claim now follows. Note that the case $s\leq-d$ is only relevant when $d=1$ given that $s\in(-2,d)$. ∎

4.4. Proof of Corollary 2.3

Let $(x_{n})_{n\in\mathbb{N}}$ be the greedy sequence of points on $\mathbb{S}^{d}$ with respect to $K_{s}$ as defined in (2.1). Arguing as in the proof of Theorem 2.2,

[TABLE]

We know from Theorem 2.2 that

[TABLE]

We also recall that

[TABLE]

We can now introduce, for $X>0$ and $N\in\mathbb{N}$, the set

[TABLE]

Collecting all these estimates, we see, for some $\alpha_{s,d}>0$,

[TABLE]

From this we deduce that

[TABLE]

which is less than $\varepsilon N$ for $X$ large enough. This proves Corollary 2.3.

5. $L^{2}$-discrepancy: Proof of Theorem 2.4

We turn to showing our main result on the $L^{2}$-spherical cap discrepancy of the greedy sequence. The relevance of the greedy construction to discrepancy is based on the following classical result [91] that relates the sum of pairwise Euclidean distances (in other words, the Riesz energy with $s=-1$) to the $L^{2}$-discrepancy.

Theorem 5.1** (Stolarsky Invariance Principle).**

For $\omega_{N,d}=\left\{x_{1},\dots,x_{N}\right\}\subset\mathbb{S}^{d}$,

[TABLE]

where the constant $c_{d}$ is given by

[TABLE]

Note that the construction of the greedy sequence in (2.1) when $s=-1$ aims to maximize the pairwise sum of Euclidean distances at every step and thus minimize the $L^{2}$-discrepancy at every step by the Stolarsky invariance principle. As will become clear in our proof of the $N^{-1/2}$ upper bound below, it is not of tremendous importance that one picks the maximum to show such an upper bound – one really only cares about having a large value in the sum. Obviously, the larger the value the better (and thus the maximum is optimal at that step) but taking values close to the maximum should suffice (and does in practice).

Proof of Theorem 2.4.

Let us assume $\left\{x_{1},\dots,x_{m}\right\}\subset\mathbb{S}^{d}$ is given and, for all $n\geq m$,

[TABLE]

We have the trivial bound

[TABLE]

and, for $n\geq m$, that

[TABLE]

Iterating this inequality, we infer, for all $n>m$, that

[TABLE]

This now implies that for the first $N$ elements

[TABLE]

from which we deduce that

[TABLE]

In the case of $d=2$, we have

[TABLE]

from which we deduce

[TABLE]

∎

Note that the result is sharp (up to lower order terms) for sequences satisfying

[TABLE]

It is clear that many such sequences exist: all involved functions are continuous so there are always points where they attain their average value.

Remark 5.2*.*

The above result, in a different form, has more or less appeared in the literature previously: for example, the result follows directly from Theorem 3.1 (and Remark 3.2) in [63]. The purpose of reproving it here is to state the result in the form of an $L^{2}$-discrepancy bound and to give more explicit constants.

5.1. Proof of Proposition 2.2

We conclude with a proof of Proposition 2.2. To establish a nontrivial lower bound, we show that the average growth of any consecutive pairs of points is bounded from below. Suppose $\left\{x_{1},\dots,x_{n}\right\}\subset\mathbb{S}^{2}$. Recall that $-\mathcal{I}_{-1,2}=\frac{4}{3}$. Fix $X\geq 0$ such that

[TABLE]

Assume that $X<\frac{4n}{3}$. In this case it is easy to see that

[TABLE]

Indeed, the average value of the function $\sum_{i=1}^{n}\|x-x_{i}\|$ is $4n/3$ and its maximum is $4n/3+X$, therefore, denoting the set above by $\Omega$, we have

[TABLE]

which implies that $\sigma_{d}(\Omega)<1/2$. Thus every hemisphere contains points $x$ satisfying the inequality $\sum_{i=1}^{n}\|x-x_{i}\|\geq 4n/3-X$.

By restricting $x$ to the hemisphere centered at $-x_{n+1}$ we conclude that

[TABLE]

Combining this with the definition of $X$, we see that

[TABLE]

If $X\geq 4n/3$, one can see immediately that (5.1) holds with the right-hand side of $4n/3-1/6>1$. Therefore, the sum $\sum_{n=1}^{N-1}\left(\max_{x\in\mathbb{S}^{2}}\sum_{i=1}^{n}\|x-x_{i}\|-\frac{4n}{3}\right)$ in (2.18) grows at least linearly in $N$, which leads to the lower bound in Proposition 2.2. The upper bound follows from the Stolarsky principle which implies

[TABLE]

and thus, for any set of points $\left\{x_{1},\dots,x_{N}\right\}\subset\mathbb{S}^{2}$,

[TABLE]

For a greedy sequence, we can write

[TABLE]

and thus

[TABLE]

Note that this upper bound (with a worse constant) could be obtained directly from the lower bound in (2.4) of Theorem 2.1. We also remark that the same arguments would also work in higher dimensions (with different constants).

6. Numerical Examples and Comments

This section contains some basic numerical examples and general comments. First we note that, in the greedy construction, finding the exact point

[TABLE]

is computationally nontrivial.

If the success of the greedy construction were to depend very strongly on finding exact maxima in each step, it would not be a very useful construction to begin with (indeed, as also indicated by the proof of Theorem 2.4, luckily this does not seem to be the case). Numerical experiments suggest the exact opposite: these types of sequences tend to be incredibly robust and seem to lead to good results even if sometimes adversarial points are added manually. Throughout this section, we consider approximate sequences obtained as follows: given $\left\{x_{1},\dots,x_{N}\right\}\subset\mathbb{S}^{d}$, we consider 100 random points and add the one maximizing the sum of distances among those. Figure 1 shows an example of a set of $N=500$ points on $\mathbb{S}^{2}$ obtained from a single individual point. We observe that the sequence looks somewhat random but avoids clusters of points more so than an actually random sequence would.

When plotting the $L^{2}-$spherical cap discrepancy of this sequence one observes, numerically, that the discrepancy seems to be relatively close to $\sim 0.5\cdot N^{-3/4}$. Note that our construction of the sequence was based on the approximation by random points and it stands to reason that Figure 2 serves as an upper bound of the true behavior of a greedy sequence. The greedy method is agnostic to what happened in the past and works well with any arbitrary initial set. We illustrate this with a simple example where we first take 250 points uniformly at random and then compute another 250 points greedily.

The result is striking: for the first 250 elements, we see that $N^{1/2}\cdot D_{L^{2},\tiny\mbox{cap}}(\omega_{N})$ is approximately constant (as expected for random points). After that there is a pronounced decay. Perhaps even more striking is that $N^{3/4}\cdot D_{L^{2},\tiny\mbox{cap}}(\omega_{N})$ is first increasing (roughly at rate $\sim N^{1/4}$ as we expect) and then quickly decreases and returns to a constant slightly above $1/2$ (see also Fig. 2).

One could wonder about higher dimensions: one would expect optimal sequences to behave as $D_{L^{2},\tiny\mbox{cap}}(\omega_{N,3})\sim N^{-2/3}$ on $\mathbb{S}^{3}$ as well as $D_{L^{2},\tiny\mbox{cap}}(\omega_{N,4})\sim N^{-5/8}$ on $\mathbb{S}^{4}$. The same basic numerical experiment leads to results compatible with this interpretation. We emphasize that, due to increasing computational cost, these experiments were carried out only for rather small values of $N\leq 500$. At this scale, small powers of $N$ or logarithms are not always easy to detect. Moreover, the effectiveness our numerical procedure of picking 100 points at random and then adding the one with the largest distance depends on the profile of the function: if large deviations occur on sets of small measure (an effective that could conceivably become more pronounced in higher dimensions), then this method will lose effectiveness. Regardless, we believe these preliminary results to be rather interesting and hope they will inspire subsequent work on the regularity of greedy sequences on $\mathbb{S}^{d}$.

We conclude with a conjecture which collects our observations in this section and speculates that the greedy sequence has nearly optimal spherical cap discrepancy.

Conjecture**.**

Let $\omega_{N,d}\subseteq\mathbb{S}^{d}$ be the first $N$ elements of the greedy sequence with respect to $K_{-1}$ as defined in (2.1). Then, for some $c=c(d)\geq 0$,

[TABLE]

7. Appendix

7.1. Gegenbauer coefficients

In this section we compute the Gegenbauer coefficients of the functions $f_{s}$ (the Riesz kernel) and $f_{s,\varepsilon}$ (its approximation) which were defined in §4.1. Various computations for such coefficients for some ranges of $s$ can be found scattered in the literature (e.g. [23, 37]). However, since we need finer properties of these coefficients (asymptotics, monotonicity, positivity) in the full range, we present detailed computations here. We proceed first with finding the Gegenbauer coefficients for the functions $f_{s,\varepsilon}$.

Lemma 7.1**.**

For $0<s$, $n\in\mathbb{N}$

[TABLE]

where $\sideset{{}_{2}}{{}_{1}}{\operatorname{F}}$ is the ordinary hypergeometric function.

Proof.

On multiple occasions, we will use the identity

[TABLE]

We shall also use the Rodrigues formula

[TABLE]

Using (7.2) with (4.3), we have, for $0<s<d$ and $\varepsilon>0$

[TABLE]

giving us the first expression in the claim. For the second one,

[TABLE]

∎

Corollary 7.1**.**

For $0<s$, $n\in\mathbb{N}_{0}$, the Gegenbauer coefficients $\widehat{f_{s,\varepsilon}}(n,d)$ are positive and decreasing in $n$.

Proof.

For $0<s<d$ and $n\in\mathbb{N}_{0}$, we see that all the summands of

[TABLE]

are positive, so $\widehat{f_{s,\varepsilon}}(n,d)>0$. We also see that for $k\in\mathbb{N}_{0}$,

[TABLE]

so the summands in (7.3) are decreasing as a function of $n$. Thus, $\widehat{f_{s,\varepsilon}}(n,d)$ is indeed a decreasing function in $n$, for $0<s<d$ and any $\varepsilon>0$. ∎

We know consider the Gegenbauer coefficients of the Riesz kernels themselves.

Lemma 7.2**.**

For $-2<s<d$, $n\in\mathbb{N}$

[TABLE]

Proof.

We again use the Rodrigues formula along with integration by parts to find, for $s\in(-2,0)\cup(0,d)$, and $n\in\mathbb{N}_{0}$,

[TABLE]

For $s=0$ and $n\geq 1$, we have a similar result:

[TABLE]

∎

The coefficients for $n\in\mathbb{N}$ are thus clearly positive, and a comparison of consecutive coefficients in all cases also shows that $\widehat{f_{s}}(n,d)$ is also decreasing as a function of $n$. A quick asymptotic analysis give us the following:

Corollary 7.2**.**

For $-2<s<d$, $n\in\mathbb{N}$, $\widehat{f_{s}}(n,d)$ is positive and decreasing as a function of $n$. Moreover

[TABLE]

We note that in the case that $d=1$, we actually have the Chebyshev polynomials of the first type instead of the Gegenbauer polynomials. These are given by $T_{0}(t)=1$ and, for $n\in\mathbb{N}$,

[TABLE]

Through this limit, one can quickly find that Lemmas 7.1 and 7.2 as well as Corollaries 7.2 and 7.1 still hold in this case.

7.2. Limits for polarization

We define, for any symmetric, lower semi-continuous kernel $K:\mathbb{S}^{d}\times\mathbb{S}^{d}\rightarrow(-\infty,\infty]$ and finite Borel measure $\mu$, the potential

[TABLE]

We define the polarization of $\mu$ to be

[TABLE]

Lemma 7.3**.**

Let $K:\mathbb{S}^{d}\times\mathbb{S}^{d}\rightarrow(-\infty,\infty]$ be a symmetric, lower semi-continuous kernel. Let $\mu$ be a finite Borel measure on $\Omega$ and $(K_{m})_{m=1}^{\infty}$ be a sequence of symmetric, continuous kernels, increasing pointwise in $m$ to $K$. Then

[TABLE]

Proof.

By the Monotone Convergence Theorem, we have that for all $x\in\mathbb{S}^{d}$,

[TABLE]

For $m\in\mathbb{N}$, let $x_{m}=\operatorname*{\operatorname{argmin}}U_{K_{m}}^{\mu}(x)$ and let $x_{\infty}=\operatorname*{\operatorname{argmin}}U_{K}^{\mu}(x)$. Thus

[TABLE]

For all $m\in\mathbb{N}$, we see that

[TABLE]

so $U_{K_{m}}^{\mu}(x_{m})$ is an increasing sequence, bounded from above by $U_{K}^{\mu}(x_{\infty})$.

Since $\mathbb{S}^{d}$ is compact, there is a convergent subsequence $x_{m_{k}}$ of $x_{m}$, with limit point $x^{*}\in\mathbb{S}^{d}$. Now, for each $j,k,l\in\mathbb{N}$ such that $j\leq k\leq l$ we know

[TABLE]

so for all $j\leq k$

[TABLE]

Thus, by continuity, for $j\in\mathbb{N}$,

[TABLE]

Thus, by the Monotone Convergence Theorem, we have

[TABLE]

Our claim now follows from (7.10), (7.11), and the fact that $U_{K}^{\mu}(x_{\infty})\leq U_{K}^{\mu}(x^{*})$. ∎

Taking $\mu=\sum_{j=1}^{N}\delta_{x_{j}}$ gives a discrete version of the result. We are unaware of a proof like this for any other polarization problems, and would like to point out that it should hold on arbitrary compact metric measure spaces as well.

8. Acknowledgements

D. Bilyk and M. Mastrianni have been supported by the NSF grant DMS-2054606. R.W. Matzke was supported by the Austrian Science Fund FWF project F5503 part of the Special Research Program (SFB) “Quasi-Monte Carlo Methods: Theory and Applications” and NSF Postdoctoral Fellowship Grant 2202877. S. Steinerberger is supported by the NSF (DMS-2123224) and the Alfred P. Sloan Foundation.