# Point Sets on the Sphere \(\mathbb{S}^{2}\) with Small Spherical Cap Discrepancy

- 340 Downloads
- 3 Citations

## Abstract

In this paper we study the geometric discrepancy of explicit constructions of uniformly distributed points on the two-dimensional unit sphere. We show that the spherical cap discrepancy of random point sets, of spherical digital nets and of spherical Fibonacci lattices converges with order *N* ^{−1/2}. Such point sets are therefore useful for numerical integration and other computational simulations. The proof uses an area-preserving Lambert map. A detailed analysis of the level curves and sets of the pre-images of spherical caps under this map is given.

## Keywords

Discrepancy Isotropic discrepancy Lambert map Level curve Level set Numerical integration Quasi-Monte Carlo Spherical cap discrepancy## 1 Introduction

Let \(\mathbb{S}^{2} = \{ \boldsymbol{z}\in\mathbb{R}^{3}: \| \boldsymbol{z}\| = 1 \}\) be the unit sphere in the Euclidean space ℝ^{3} provided with the norm ∥⋅∥ induced by the usual inner product * x*⋅

*. On this sphere we consider the Lebesgue surface area measure*

**y***σ*normalised to a probability measure (\(\int_{\mathbb{S}^{2}} \mathrm {d}\sigma= 1\)).

*N*grows. Given a triangular scheme {

**z**_{1,N },…,

**z**_{ N,N }},

*N*≥1, of points on \(\mathbb{S}^{2}\) in such a case one has

*spherical cap discrepancy*.

*C*=

*C*(

*,*

**w***t*) centred at \(\boldsymbol {w}\in \mathbb{S}^{2}\) with height

*t*∈[−1,1] is given by the set

*C*(

*,*

**w***t*) then is

Let \(Z_{N} = \{\boldsymbol{z}_{0}, \ldots, \boldsymbol{z}_{N-1}\} \subseteq\mathbb{S}^{2}\) be an *N*-point set on the sphere \(\mathbb{S}^{2}\). The local discrepancy with respect to a spherical cap *C* measures the difference between the proportion of points in *C* (the empirical measure of *C*) and the normalised surface area of *C*. The spherical cap discrepancy is then the supremum of the local discrepancy over all spherical caps, as stated in the following definition.

## Definition 1

*spherical cap discrepancy*of an

*N*-point set \(Z_{N} = \{ \boldsymbol{z}_{0},\ldots, \boldsymbol{z}_{N-1}\} \subseteq\mathbb{S}^{2}\) is

*Z*

_{ N }is well-distributed, then this discrepancy is small. In fact, a sequence of

*N*-point systems (

*Z*

_{ N })

_{ N≥1}satisfying

*asymptotically uniformly distributed*. Using, for example, the classical

*Erdös–Turán type inequality*(cf. Grabner [25], also cf. Li and Vaaler [38]) or

*LeVeque type inequalities*(Narcowich et al. [41]) and the fact that the set of polynomials is dense in the set of continuous functions, one can show that (2) is equivalent to (1b).

*c*,

*C*>0, independent of

*N*, such that a

*low-discrepancy*scheme \(\{ Z_{N}^{*} \}_{N \geq2}\) satisfies

*N*-point sets

*Z*

_{ N }on \(\mathbb{S}^{2}\) and there always exists an

*N*-point set \(Z_{N} \subseteq\mathbb{S}^{2}\) such that the upper bound holds. The proof of the upper bound is probabilistic in nature and is thus non-constructive. To our best knowledge, explicit constructions of low-discrepancy schemes are not known. (In this paper we restrict ourselves to the sphere \(\mathbb{S}^{2}\), though some of the results are known for spheres of dimension

*d*≥2.)

*Z*

_{ N }with small spherical cap discrepancy has been given in [39, 40]. For instance, in [39] it was shown that

*Z*

_{ N }for which we have

*c*≤1, see Tables 1, 2 below.

Comparison of the effect of two different normalisations of the value \(\widetilde{D}(Z_{N})\) computing \(\max_{\boldsymbol{w}\in Z_{N}} \delta(Z_{N}; \boldsymbol{w})\), cf. Definition 1, for a digital net *Z* _{ N } based on a two-dimensional Sobol’ point set

| 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |

| 64 | 128 | 256 | 512 | 1024 | 2048 | 4096 | 8192 |

\(\frac{N^{3/4}\widetilde{D}(Z_{N})}{\sqrt{\log N}}\) | 0.8829 | 0.8436 | 0.8279 | 0.8632 | 0.8518 | 1.2128 | 1.2285 | 0.9546 |

\(\frac{N^{3/4} \widetilde{D}(Z_{N})}{\log N}\) | 0.4329 | 0.3829 | 0.3515 | 0.3456 | 0.3235 | 0.4392 | 0.4259 | 0.3180 |

| 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 |

| 16384 | 32768 | 65536 | 131072 | 262144 | 524288 | 1048576 | 2097152 |

\(\frac{N^{3/4} \widetilde {D}(Z_{N})}{\sqrt{\log N}}\) | 0.7925 | 0.8862 | 1.0331 | 0.8337 | 0.8562 | 0.9854 | 1.1167 | 1.1463 |

\(\frac{N^{3/4} \widetilde{D}(Z_{N})}{\log N}\) | 0.2544 | 0.2748 | 0.3102 | 0.2428 | 0.2424 | 0.2715 | 0.2999 | 0.3004 |

Comparison of the effect of two different normalisations of the value \(\widetilde{D}(Z_{F_{m}})\) computing \(\max_{\boldsymbol {w}\in Z_{F_{m}}} \delta( Z_{F_{m}}; \boldsymbol{w})\), cf. Definition 1, for spherical Fibonacci points \(Z_{F_{m}}\)

| 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

| 1597 | 2584 | 4181 | 6765 | 10946 | 17711 | 28657 | 46368 |

\(\frac{F_{m}^{3/4} \widetilde{D}(Z_{F_{m}})}{\sqrt{\log F_{m}}}\) | 0.6729 | 0.6373 | 0.6228 | 0.6661 | 0.6953 | 0.6890 | 0.7427 | 0.6900 |

\(\frac{F_{m}^{3/4} \widetilde {D}(Z_{F_{m}})}{\log F_{m}}\) | 0.2477 | 0.2273 | 0.2156 | 0.2243 | 0.2279 | 0.2203 | 0.2318 | 0.2105 |

| 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 |

| 75025 | 121393 | 196418 | 317811 | 514229 | 832040 | 1346269 | 2178309 |

\(\frac{F_{m}^{3/4} \widetilde {D}(Z_{F_{m}})}{\sqrt{\log F_{m}}}\) | 0.6957 | 0.7249 | 0.7531 | 0.7205 | 0.8562 | 0.7455 | 0.7862 | 0.8082 |

\(\frac{F_{m}^{3/4} \widetilde{D}(Z_{F_{m}})}{\log F_{m}}\) | 0.2076 | 0.2118 | 0.2157 | 0.2024 | 0.2361 | 0.2019 | 0.2092 | 0.2115 |

*spherical cap*\(\mathbb{L}_{2}\)-

*discrepancy*

*sum of distances*and its continuous counterpart the

*distance integral*by means of

*Stolarsky’s Invariance Principle*[50] for the Euclidean distance and the 2-sphere: This gives a simple way of computing the spherical cap \(\mathbb {L}_{2}\)-discrepancy of point sets on \(\mathbb{S}^{2}\). In [14] it is shown that the spherical cap \(\mathbb{L}_{2}\)-discrepancy of

*Z*

_{ N }can be interpreted as the worst-case error of an equal-weight numerical integration rule with node set

*Z*

_{ N }for functions in the unit ball of a certain Sobolev space over \(\mathbb{S}^{2}\). It is shown in [13] that on average (i.e., for randomly chosen points independently identically uniformly distributed over the sphere), the expected squared worst-case error is of the form (4/3)

*N*

^{−1}. Thus the expected value of the squared spherical cap discrepancy satisfies

*typical*asymptotic order of the spherical cap discrepancy of random point sets in detail in Sect. 4. Among other results, we show that there is also a constant

*C*>0 such that \(\mathbb{E} [ D(Z_{N}) ] \le C N^{-1/2}\).

*c*>0, not depending on

*N*,

*Coulomb potential energy*essentially given by

*K*-

*regular*test sets

^{1}introduced by Sjögren [46], the estimate above is sharp in the following sense: The upper bound holds for any

*K*-regular test set, whereas there are some numbers

*K*

_{0}and

*c*such that to any

*N*points \(\boldsymbol{z}_{1}, \dots, \boldsymbol{z}_{N} \in\mathbb {S}^{2}\) there is a

*K*

_{0}-regular test set

*B*with [24, Corollary 2]

*Spherical*

*n-designs*introduced by Delsarte, Goethals and Seidel in the landmark paper [19] are node sets for equal-weight numerical integration rules such that all spherical polynomials of degree ≤

*n*are integrated exactly. Grabner and Tichy [26] give the following upper bound of the spherical cap discrepancy of a spherical

*n*-design with

*N*(

*n*) points:

*Erdös–Turán type inequality*. (See also Andrievskii et al. [4] for a similar form for

*K*-regular test sets.)

A spherical *n*-design is the solution of a system of polynomial equations (one for every spherical harmonic of the real orthonormal basis of the space of spherical polynomials of degree ≤*n*). Hence, a natural lower bound for the number of points of a spherical *n*-design is given by the dimension of the involved polynomial space; that is, one needs at least ≥*n* ^{2}/4 points. The famous conjecture that *C*
*n* ^{2} points (for some universal *C*>0) are sufficient for a spherical *n*-design seems to have been settled by Bondarenko et al. [10]. The proposed proof is non-constructive. Hardin and Sloane [29] propose a construction of so-called putative spherical *n*-designs with (1/2)*n* ^{2}+*o*(*n* ^{2}) points. The variational characterisation of spherical designs introduced in [48] (also cf. [26]) leads to a minimisation problem for a certain energy functional (changing with *n*) whose minimiser is a spherical *n*-design if and only if the functional becomes zero. Numerical results also suggest a coefficient 1/2. When allowing more points, *N*(*n*)=(*n*+1)^{2}, interval-based methods yield, in principle, the existence of a spherical *n*-design near so-called extremal (maximum determinant points, cf. [47]). Due to the computational cost this approach was carried out only for *n*≤20. Very recently, Chen et al. [16] devised a computational algorithm based on interval arithmetic that, upon successful completion, verifies the existence of a spherical *n*-design with (*n*+1)^{2} points and provides narrow interval enclosures which are known to contain these nodes with mathematical certainty. The spherical cap discrepancy of all such obtained spherical *n*-design with \(\mathcal{O}(n^{2})\) points can then be bounded by *C*′*N* ^{−1/2} by (6). For the sake of completeness it should be mentioned that the tensor product rules used by Korevaar and Meyers [33] to prove the existence of spherical *n*-designs of \(N(n) = \mathcal {O}(n^{3})\) points give rise to *N*(*n*)-point configurations whose spherical cap discrepancy can be bounded by *C*′′[*N*(*n*)]^{−1/3} by (6).

From [14] it follows that the spherical cap discrepancy of a point set \(Z_{N} = \{\boldsymbol{z}_{0},\ldots, \boldsymbol {z}_{N-1}\} \subseteq \mathbb{S}^{2}\) yields an upper bound on the integration error in certain Sobolev spaces of functions defined on \(\mathbb{S}^{2}\) using a quadrature rule \(Q_{N}(f) = \frac{1}{N} \sum_{n=0}^{N-1} f(\boldsymbol{z}_{n})\). Thus, our results here provide an explicit mean of finding quadrature points for numerical integration of functions defined on \(\mathbb{S}^{2}\). Our result here improves the bound on the integration error in [12] by a factor of \(\sqrt{\log N}\).

^{2}to \(\mathbb{S}^{2}\) using an equal area transformation \(\boldsymbol{\varPhi}:[0,1]^{2} \to \mathbb{S}^{2}\). The same approach has previously been used in [12] and [27], in both cases in the context of numerical integration. The low-discrepancy points in [0,1]

^{2}are obtained from digital nets and Fibonacci lattices, see [20, 42]. These point sets are well-distributed with respect to rectangles anchored at the origin (0,0). However, the set

Hence, in order to prove bounds on the spherical cap discrepancy of digital nets and Fibonacci lattices lifted to the sphere using * Φ* (we call those point sets spherical digital nets and spherical Fibonacci lattices), we need to prove bounds on a general notion of discrepancy in [0,1]

^{2}. To this end we study discrepancy in [0,1]

^{2}with respect to convex sets, the corresponding discrepancy being known as

*isotropic discrepancy*[8]. We show that digital nets and Fibonacci lattices have isotropic discrepancy of order \(\mathcal{O}(N^{-1/2})\). Using these result and some properties of the function

*, we can show that spherical digital nets and spherical Fibonacci lattices have spherical cap discrepancy at most*

**Φ***CN*

^{−1/2}for an explicitly given constant

*C*, see Corollaries 16 and 18. Note that the best possible rate of convergence of the isotropic discrepancy is

*N*

^{−2/3}(log

*N*)

^{ c }for some 0≤

*c*≤4, see [7] and [8, p. 107]. Hence the approach via the isotropic discrepancy cannot give the optimal rate of convergence for the spherical cap discrepancy.

In the following we define the equal area Lambert map * Φ* and show some of its properties.

## 2 The Equal-Area Lambert Transform and Some Properties

*Lambert cylindrical equal-area projection*

The area-preserving Lambert map can be illustrated in the following way. The unit square [0,1]^{2} is linearly stretched to the rectangle [0,2*π*]×[−1,1], rolled into a cylinder of radius 1 and height 2 and fitted around the unit sphere such that the polar axis is the main *z*-axis. This way a point (*α*,*τ*) in [0,1]^{2} is mapped to a point on the cylinder which is radially projected along a ray orthogonal to the polar axis onto the sphere giving the point * Φ*(

*α*,

*τ*).

*with height*

**w***t*under the Lambert map is the set

The sets *B*(* w*,

*t*) are not convex, in general. Thus, we consider a more general class of sets which we call pseudo-convex. A definition is given in the following.

## Definition 2

Let *A* be an open subset of [0,1]^{2} such that there exists a collection of *p* convex subsets *A* _{1},…,*A* _{ p } of [0,1]^{2} with the following properties: (a) *A* _{ j }∩*A* _{ k } is empty for *j*≠*k*; (b) *A*⊆*A* _{1}∪⋯∪*A* _{ p }; (c) either *A* _{ j } is a convex part of *A* (*A* _{ j }⊆*A*) or the complement of *A* with respect to *A* _{ j }, \(A_{j}^{\prime}= A_{j} \setminus A\), is convex. Then *A* is called a *pseudo-convex* set and *A* _{1},…,*A* _{ p } is an admissible convex covering for *A* with *p* parts (with *q* convex parts of *A*).

## Lemma 3

*For every* \(\boldsymbol{w}\in\mathbb{S}^{2}\) *and all* −1≤*t*≤1, *the pre*-*image* *B*(* w*,

*t*)

*of the spherical cap*

*C*(

*,*

**w***t*)

*centred at*

**w***with height*

*t*

*under the Lambert map is pseudo*-

*convex with an admissible convex covering with at most*7

*parts*.

*More precisely*,

*taking into account the number of convex parts of the pre*-

*image*,

*among the convex coverings with*

*p*

*parts and*

*q*

*of which are convex*,

*the worst case has*

*p*=7

*and*

*q*=3

*which implies the constant*2

*p*−

*q*=11.

The proof of Lemma 3 in Sect. 7 gives details how to construct admissible coverings.

## 3 Isotropic- and Spherical Cap Discrepancy

We introduce the isotropic discrepancy of a point set and a sequence as follows. Let *λ* be the Lebesgue area measure in the unit square.

## Definition 4

*isotropic discrepancy*

*J*

_{ N }of an

*N*-point set

*P*

_{ N }={

**x**_{0},…,

**x**_{ N−1}} in [0,1]

^{2}is defined as

^{2}.

For an infinite sequence **x**_{0},**x**_{1},…∈[0,1]^{2} the isotropic discrepancy is defined as the isotropic discrepancy of the initial *N* points of the sequence.

## Lemma 5

*Let*

*A*

*be a pseudo*-

*convex subset of*[0,1]

^{2}

*with an admissible convex covering of*

*p*

*parts with*

*q*

*convex parts of*

*A*.

*Then for any*

*N*-

*point set*

*P*

_{ N }={

**x**_{0},…,

**x**_{ N−1}}⊆[0,1]

^{2},

## Proof

*A*

_{1},…,

*A*

_{ p }be an admissible convex covering of

*A*with

*p*parts. Without loss of generality, let

*A*

_{1},…,

*A*

_{ q }be the convex parts of

*A*and

*A*

_{ q+1},…,

*A*

_{ p }those for which \(A_{j}^{\prime}= A_{j} \setminus A\) (

*q*+1≤

*j*≤

*p*) is convex. Clearly,

## Theorem 6

*Let*

*P*

_{ N }={

**x**_{0},…,

**x**_{ N−1}}⊆[0,1]

^{2}

*and let*\(Z_{N} = \{\boldsymbol{\varPhi}(\boldsymbol{x}_{0}),\ldots, \boldsymbol {\varPhi}(\boldsymbol{x}_{N-1})\} \subseteq \mathbb {S}^{2}\).

*Then*

## Proof

*t*≤1. A point

*(*

**Φ**

**x**_{ n })∈

*C*(

*,*

**w***t*) if and only if

**x**_{ n }∈

*B*(

*,*

**w***t*). Thus,

*preserves areas, we have*

**Φ***p*−

*q*=11 from Lemma 3 we arrive at the result. □

We have now reduced the problem of proving bounds on the spherical cap discrepancy to prove bounds on the isotropic discrepancy of points in the square [0,1]^{2}. We will study this problem in Sect. 5.

## 4 Spherical Cap Discrepancy of Random Points Sets

*P*be a probability on it. Let further

*X*

_{ n },

*n*≥0, denote a sequence of independent, identically distributed (i.i.d.) random variables on a probability space \((\varOmega, \mathcal{A}, \mathbb{P})\) with values in

*M*, and let \(\mathcal{C} \subseteq\mathcal{M}\) denote a class of subsets of

*M*. To avoid measurability problems we will assume throughout the rest of this section that the class \(\mathcal{C}\) is countable. Let

*A*⊆

*M*be an arbitrary set. Then \(\mathcal{C}\) is said to

*shatter*

*A*if to every possible subset

*B*of

*A*there exists a set \(C \in\mathcal{C}\) such that

*k*≥1 the

*k*th

*shattering coefficient*\(S_{\mathcal {C}}(k)\) of \(\mathcal{C}\) is defined as

*Vapnik–Červonenkis dimension*(VC-dimension) of \(\mathcal {C}\) is defined as

*Vapnik–Červonenkis class*(VC class). The theory of VC classes is of extraordinary importance in the theory of empirical processes indexed by classes of functions. For example, a class \(\mathcal{C}\) is

*uniformly Glivenko–Cantelli*if and only if it is a VC class, see [53]. We will use the following theorem, which is a combination of results of Talagrand [51, Theorem 6.6] and Haussler [30, Corollary 1], and has already been used by Heinrich et al. [31] in the context of probabilistic discrepancy theory.

## Theorem 7

(See [31, Theorem 2])

*There exists a positive number*

*K*

*such that*,

*for each VC class*\(\mathcal{C}\)

*and each probability*

*P*

*and sequence*

*X*

_{ n },

*n*≥0,

*as above*,

*the following holds*:

*For all*\(s \geq K \sqrt{v (\mathcal {C})}\)

*we have*

In our setting we will have \(M=\mathbb{S}^{2}\), \(\mathcal{M}\) will denote the sigma-field generated by the class of spherical caps, *P* will stand for the normalised Lebesgue surface area measure *σ*, and \(\mathcal{C}\) will denote the class of all spherical caps for which the centre * w* is a vector of rational numbers and the height

*t*is also a rational number (this restriction is necessary to assure that the class \(\mathcal{C}\) is countable; of course, the spherical cap discrepancy with respect to this class is the same as the discrepancy with respect to the class of

*all*spherical caps). In the sequel we assume that the i.i.d. random variables

*X*

_{ n },

*n*≥0, are uniformly distributed on \(\mathbb{S}^{2}\). We will write

*Z*

_{ N }=

*Z*

_{ N }(

*ω*) for the (random) point set {

*X*

_{0},…,

*X*

_{ N−1}}={

*X*

_{0}(

*ω*),…,

*X*

_{ N−1}(

*ω*)}.

The following proposition asserts that the class \(\mathcal{C}\) is a VC class (the proof of this and the subsequent results of this section can be found in Sect. 7).

## Proposition 8

*The class* \(\mathcal{C}\) *has VC dimension* 5.

Using Theorem 7 and Proposition 8 we can prove the following results:

## Theorem 9

*There exist constants*

*C*

_{1},

*C*

_{2}

*such that for*

*N*≥1,

## Remark

The existence of such a constant *C* _{1} for the lower bound follows directly from (5); we can choose *C* _{1}=6^{−1/2}.

## Theorem 10

*For any*

*ε*>0

*there exist positive constants*

*C*

_{3}(

*ε*),

*C*

_{4}(

*ε*)

*such that for sufficiently large*

*N*,

*typical*discrepancy of a random set of

*N*points is of order

*N*

^{−1/2}. However, actually much more is true, since by classical results any VC class \(\hat{\mathcal{C}}\) on a measurable space \((\hat{M},\hat{\mathcal{M}})\) is a so-called

*Donsker class*, which essentially means that for every probability measure \(\hat{P}\) and every sequence

*V*

_{ n },

*n*≥0, of i.i.d. random variables having law \(\hat{P}\) the

*empirical process indexed by sets*

*B*(

*C*), which has covariance structure

*N*→∞; however, to keep this presentation short and self-contained we will not pursue this method any further, and refer the interested reader to [2, 21, 22, 52] and the references therein.

## Remark

The upper bounds in Theorems 9 and 10 follow from Theorem 7. However, since no concrete value for the constant *K* in Theorem 7 is known, the value of the constants *C* _{2} and *C* _{4} in Theorem 9 and Theorem 10, respectively, is also unknown. It is possible that the decomposition technique from [1] can be used to achieve a version of Theorems 9 and 10 with explicitly known constants in the upper bound.

Finally, the following theorem describes the asymptotic order of a *typical* infinite sequence of random points.

## Theorem 11

*We have*

*bounded law of the iterated logarithm*, and follows easily from Theorem 6 and Philipp’s law of the iterated logarithm (LIL) for the isotropic discrepancy of random point sets in the plane. More precisely, Philipp [44] proved that for a sequence of i.i.d. uniformly distributed random variables

*Y*

_{ n },

*n*≥0, on the unit square (writing

*P*

_{ N }for the (random) point set {

*Y*

_{0},…,

*Y*

_{ N−1}}), the law of the iterated logarithm

*C*

^{∗}denote a fixed spherical cap with area 2

*π*(which means that

*C*

^{∗}is a hemisphere, and has normalised surface area measure

*σ*(

*C*

^{∗})=1/2). Then clearly the random variables

## 5 Point Sets with Small Isotropic Discrepancy

In this section we investigate the isotropic discrepancy of (0,*m*,2)-nets and Fibonacci lattices. In particular we show that the isotropic discrepancy of those point sets converges with order \(\mathcal {O}(N^{-1/2})\). Note that the best possible rate of convergence of the isotropic discrepancy is *N* ^{−2/3}(log*N*)^{ c } for some 0≤*c*≤4, see [7] and [8, p. 107]. Whether (0,*m*,2)-nets and/or Fibonacci lattices achieve the optimal rate of convergence for the isotropic discrepancy is an open question.

### 5.1 Nets and Sequences

We give the definition of (0,*m*,2)-nets in base *b* in the following.

## Definition 12

*b*≥2 and

*m*≥1 be integers. A point set \(P_{b^{m}} \subseteq[0,1)^{2}\) consisting of

*b*

^{ m }points is called a (0,

*m*,2)-

*net in base*

*b*, if for all nonnegative integers

*d*

_{1},

*d*

_{2}with

*d*

_{1}+

*d*

_{2}=

*m*, each of the elementary intervals

It is also possible to construct nested (0,*m*,2)-nets, thereby obtaining an infinite sequence of points.

## Definition 13

Let *b*≥2 be an integer. A sequence **x**_{0},**x**_{1},…∈[0,1)^{2} is called a (0,2)-*sequence in base* *b*, if for all *m*>0 and for all *k*≥0, the point set \(\boldsymbol{x}_{k b^{m}}, \boldsymbol{x}_{kb^{m} + 1}, \ldots, \boldsymbol{x}_{(k+1)b^{m}-1}\) is a (0,*m*,2)-net in base *b*.

Explicit constructions of (0,*m*,2)-nets and (0,2)-sequences are due to Sobol’ [49] and Faure [23], see also [20, Chap. 8].

The following is a special case of an unpublished result due to Gerhard Larcher. For completeness we include a proof here.

## Theorem 14

*For the isotropic discrepancy*

*J*

_{ N }

*of a*(0,

*m*,2)-

*net*

*P*

_{ N }

*in base*

*b*(

*N*=

*b*

^{ m })

*we have*

## Proof

*k*=⌊

*m*/2⌋ and consider a subcube

*W*of [0,1)

^{ s }of the form

*c*

_{ i }<

*b*

^{ k }(

*c*

_{ i }an integer) for

*i*=1,2. The cube

*W*has volume

*b*

^{−2k }and is the union of

*b*

^{ m−2k }elementary intervals of order

*m*. Indeed,

*W*contains exactly

*b*

^{ m−2k }points of the net. The diagonal of

*W*has length \(\sqrt{2} / b^{k}\).

*A*be an arbitrary convex subset of [0,1]

^{2}. Let

*W*

^{∘}denote the union of cubes

*W*fully contained in

*A*and let \(\overline{W}\) denote the union of cubes

*W*having nonempty intersection with

*A*or its boundary. The sets \(\overline{W}\) and

*W*

^{∘}are fair with respect to the net, that is,

*A*is convex, the length of the boundary of

*A*is at most the circumference of the unit square, which is 4. Further we have

^{2}). Moreover,

*A*, the boundary of

*A*has length at most 4 (which is the circumference of the square [0,1]

^{2}).

Note that the above result only applies when the number of points *N* is of the form *N*=*b* ^{ m } (notice that choosing *m*=1 only yields a trivial result, hence one usually chooses a small base *b* and a ‘large’ value of *m*). In the following we give an extension where the number of points can take on arbitrary positive integers.

## Theorem 15

*For the isotropic discrepancy*

*J*

_{ N }

*of the first*

*N*

*points*

*P*

_{ N }={

**x**_{0},…,

**x**_{ N−1}}

*of a*(0,2)-

*sequence in base*

*b*,

*we have*

## Proof

*N*∈ℕ have base

*b*expansion

*N*=

*N*

_{0}+

*N*

_{1}

*b*+⋯+

*N*

_{ m }

*b*

^{ m }. Let

*P*

_{ N }={

**x**_{0},…,

**x**_{ N−1}} denote the first

*N*points of a (0,2)-sequence in base

*b*. For 0≤

*k*≤

*m*with

*N*

_{ k }>0 and 0≤

*ℓ*<

*N*

_{ k }the point set

*k*,2)-net in base

*b*by Definition 13. Thus

*P*

_{ N }is a disjoint union of such (0,

*k*,2)-nets:

*C*we have

*a*−1)(

*a*+1)=

*a*

^{2}−1. □

## Corollary 16

- (1)
*Let**P*_{ N }*be a*(0,*m*,2)-*net in base**b**and let*\(Z_{N} = \boldsymbol{\varPhi}(P_{N}) \subseteq\mathbb{S}^{2}\).*Then the spherical cap discrepancy**D*(*Z*_{ N })*is bounded by*$$ D(Z_N) \le44 \sqrt{2} b^{-\lfloor m/2 \rfloor}. $$ - (2)
*Let**P*_{ N }*be the first**N**points of a*(0,2)-*sequence in base**b**and let*\(Z_{N} = \boldsymbol{\varPhi}(P_{N}) \subseteq\mathbb {S}^{2}\).*Then the spherical cap discrepancy**D*(*Z*_{ N })*is bounded by*$$ D(Z_N) \le44 \sqrt{2} \bigl( b^2 + b^{3/2} \bigr) \frac{1}{\sqrt{N}} \quad\mbox{\textit{for all}}\ N. $$

Note that item (2) improves upon Theorem 11 by a factor of \(\sqrt{\log\log N}\) and hence, asymptotically, spherical digital sequences are better than random sequences almost surely.

*Z*

_{ N }based on two-dimensional Sobol’ point sets by explicit numerical computation of

### 5.2 Fibonacci Lattices

*F*

_{ m }are given by

*F*

_{1}=1,

*F*

_{2}=1 and

*F*

_{ m }=

*F*

_{ m−1}+

*F*

_{ m−2}for all

*m*>2. A Fibonacci lattice is a point set of

*F*

_{ m }points in [0,1)

^{2}given by

*x*}=

*x*−⌊

*x*⌋ denotes the fractional part for nonnegative real numbers

*x*. The set

In the following we prove a bound on the isotropic discrepancy of Fibonacci lattices, see also [35, 36, 37].

## Lemma 17

*For the isotropic discrepancy*\(J_{F_{m}}\)

*of a Fibonacci lattice*\(\mathcal{F}_{m}\)

*we have*

## Proof

*m*first. From [43, Theorem 3] it follows that for

*m*∈ℕ the Fibonacci lattice \(\mathcal{F}_{2m+1}\) can be generated by the vectors This means that

*U*(

**f**_{ n }) a unit cell (belonging to the point

**f**_{ n }). Note that the area of a unit cell is 1/

*F*

_{2m+1}and each unit cell contains exactly one point of the lattice, see [43].

**a**_{2m+1}⊥

**b**_{2m+1}, it follows that the minimum distance between points of the Fibonacci lattice is

Let now *A* be an arbitrary convex subset of [0,1]^{2}. Let *W* ^{∘} denote the union of all unit cells fully contained in *A* and let \(\overline{W}\) denote the union of all unit cells with nonempty intersection with *A* or its boundary.

*A*is convex, the length of the boundary of

*A*is at most the circumference of the unit square, which is 4. Further we have

^{2}).

*A*, the boundary of

*A*has length at most 4 (which is the circumference of the square [0,1]

^{2}).

*n*=2

*m*with

*m*≥2. Using the identity

*F*

_{ m }

*F*

_{2m−1}−

*F*

_{ m−1}

*F*

_{2m }=(−1)

^{ m−1}

*F*

_{ m }, we obtain

*F*

_{ m }

*F*

_{2m−1}≡(−1)

^{ m−1}×

*F*

_{ m }(mod

*F*

_{2m }). Consequently,

*F*

_{ m+1}

*F*

_{2m−1}−

*F*

_{ m }

*F*

_{2m }=(−1)

^{ m }

*F*

_{ m−1}, we obtain the generating vector

**a**_{2m }and

**b**_{2m }is

**a**_{2m }and

**b**_{2m }does not contain any point of the Fibonacci lattice in its interior (i.e. is a unit cell of the Fibonacci lattice, see [15, 43]). Thus, we have

*U*(

**f**_{ n }) a unit cell (belonging to the point

**f**_{ n }). Note that the area of a unit cell is 1/

*F*

_{2m+1}and each unit cell contains exactly one point of the lattice.

**a**_{2m }+

**b**_{2m }∥,∥

**a**_{2m }−

**b**_{2m }∥>∥

**a**_{2m }∥. Thus the minimum distance between points of the Fibonacci lattice is

**a**_{2m }+

**b**_{2m }∥. Using the relations

*F*

_{ m }=

*F*

_{ m−1}+

*F*

_{ m−2}, \(F_{m}^{2} +F_{m-1}^{2} = F_{2m-1}\) and

*F*

_{2m }=(2

*F*

_{ m−1}+

*F*

_{ m })

*F*

_{ m }, we obtain Thus the diameter of a unit cell is bounded by

## Corollary 18

*Let*\(\mathcal{F}_{m}\)

*be a Fibonacci lattice and let*\(Z_{F_{m}} = \boldsymbol{\varPhi}(\mathcal{F}_{m}) \subseteq\mathbb{S}^{2}\).

*Then the spherical cap discrepancy*\(D(Z_{F_{m}})\)

*is bounded by*

## 6 Level Curves of the Distance Function and Their Properties

*=*

**w***(*

**Φ***u*,

*v*) and

*=*

**z***(*

**Φ***α*,

*τ*), can be written as The well-defined boundary curve of a spherical cap

*C*(

*,*

**w***t*) has the implicit representation

*to a point on the boundary of*

**w***C*(

*,*

**w***t*).) Relation (8) describes a level curve \(\mathcal {C}_{\boldsymbol{w} }\) of the distance function ∥

*−*

**w***(*

**Φ***α*,

*τ*)∥ (for

*fixed) in the parameter space which is the unit square, cf. Fig. 2. For further references we record that for each*

**w***there are, in general, two exceptional levels, where the boundary of the spherical cap centred at*

**w***passes through the North Pole (*

**w***) and the South Pole (−*

**p***), respectively, which may coincide if*

**p***is on the equator. For these level curves the singular behaviour at the poles imposed by the parameterisation*

**w***plays a role.*

**Φ**Suppose *u*=1/2. Because the sign of the difference *u*−*α* is absorbed by the cosine function in the distance function, a level curve (a level set) is symmetric with respect to the vertical line *α*=*u*. The shape of the curves (sets) does not change when the point * w* is rotated about the polar axis except a part moving outside the left side of the unit square enters at the right side (“wrap around”). This “modulo 1” behaviour complicates considerations regarding convexity of the pre-image of a spherical cap centred at

*under the distance function. Similarly and in the same sense, the level curves (sets) are symmetric with respect to the vertical line*

**w***α*=

*u*±1/2mod1 which passes through the parameter point of the antipodal point −

*=*

**w***(*

**Φ***u*±1/2mod1,1−

*v*). When identifying the left and right sides of the unit square [0,1]

^{2}, we get the “cylindrical view”. Clearly, all level curves except the critical ones associated with the distance to one of the poles are closed on the open cylinder. Thus, the critical level curves separate the open cylinder into three parts corresponding to the cases when neither pole is contained in the spherical cap centred at

*, only one pole is contained in the spherical cap, and both poles are contained in the spherical cap. In both the first and the last case the level curve cannot escape the level set bounded by a critical curve (and the boundary of the cylinder). It is closed even in the unit square provided the level set is contained in its interior. In the middle case the level curves wrap around the cylinder; that is, start at the left side of the unit square and end at its right side at the same height; cf. Fig. 2.*

**w**## Let **w** be the North or the South Pole

**w**

Then the level curves are horizontal lines in the unit square which are smooth curves. Moreover, the pre-image of a spherical cap centred at one of these poles is a convex set (a rectangle).

## Let **w** be Different from Either Pole (that is, 0<*v*<1)

**w**

We observe that the partial derivatives involving differentiation with respect to *τ* become singular as *τ* approaches 0 (North Pole) or 1 (South Pole).

*α*,

*τ*) of \(\mathcal {C}_{\boldsymbol{w} }\) is given by

*π*(

*u*−

*α*))=0; that is, either

*α*=

*u*or

*α*=

*u*±1/2mod1. In the first case one has cos(2

*π*(

*u*−

*α*))=1 and, therefore, \(F_{\alpha}^{2} + F_{\tau}^{2} = 0\) if and only if

*τ*=

*v*. In the second case one has cos(2

*π*(

*u*−

*α*))=−1 and, therefore, \(F_{\alpha}^{2} + F_{\tau}^{2} = 0\) if and only if

*τ*=1−

*v*. It follows that the denominator of the curvature formula (10) vanishes if and only if

*=*

**z***(*

**Φ***α*,

*τ*) (on the boundary of the spherical cap centred at

*w*=

*(*

**Φ***u*,

*v*)) coincides with

*(that is, the spherical cap degenerates to the point*

**w***) or*

**w***coincides with the antipodal point of*

**z***(that is, the closed spherical cap is the whole sphere). In either of these cases the pre-image of the spherical cap under*

**w***is convex.*

**Φ***spherical cap is neither a point nor the whole sphere*. Then \(F_{\alpha}^{2} + F_{\tau}^{2} \neq0\). For the numerator in (10) we obtain after some simplifications: The right-hand side above can be written as a polynomial in

*x*=cos(2

*π*(

*u*−

*α*)) as follows: where

*x*, we observe that the coefficient of

*x*

^{2}vanishes, and after simplifications we arrive at The coefficient of

*x*

^{3}does not vanish for 0<

*v*<1 and 0<

*τ*<1. Hence, we divide and get

*A*,

*B*, and

*H*)

*p*(1−

*τ*)=

*p*(

*τ*) and

*q*(1−

*τ*)=−

*q*(

*τ*). Hence,

*Q*(

*τ*;

*x*)=−

*Q*(1−

*τ*;−

*x*) for all

*x*. In particular, if

*ξ*is a zero of

*Q*(

*τ*;⋅), then so is −

*ξ*a zero of

*Q*(1−

*τ*;⋅) and vice versa. The monic polynomial

*Q*of degree 3 with real coefficients has either one or three real solutions (counting multiplicity). With the help of Mathematica we find that the discriminant of the polynomial

*Q*is positive:

*Q*has three distinct real roots.

*v*=1/2 the polynomial

*Q*reduces to

*τ*=1/2) correspond to

*(*

**Φ***α*,

*τ*)=±

*and can be discarded, since we assumed that the spherical cap is neither a point nor the whole sphere. The solution zero yields that cos(2*

**w***π*(

*u*−

*α*))=0, which in turn shows that the zeros of the curvature (10) form the vertical lines at

*α*=

*u*±1/4mod1 if

*v*=1/2.

*v*≠1/2. Suppose

*Q*has a zero at ±1. Then

*τ*=

*v*. This implies that

*(*

**Φ***α*,

*τ*)=

*, which is excluded by our assumptions. Suppose*

**w***Q*has a zero at 0. Since

*v*≠1/2, this can only happen when

*τ*=1/2.

*τ*=1/2), and 1 cannot be zeros of the polynomial

*Q*, we use Sturm’s theorem to show that the polynomial

*Q*has precisely one solution either in the interval (−1,0) or in the interval (0,1) if

*τ*≠1/2, cf. Table 3. First, we generate the canonical Sturm chain by applying Euclid’s algorithm to

*Q*and its derivative: Let

*σ*(

*x*) denote the number of sign changes (not counting a zero) in the sequence

*x*=0 we obtain the canonical Sturm chain \(\{q, p, -q, \operatorname{discr}(Q) / (4 p^{2})\}\) and we conclude that

*σ*(0)=2 for (1/2−

*v*)(1/2−

*τ*)>0 and

*σ*(0)=1 otherwise. For

*x*=1 we have (The positivity of

*p*

_{2}(1) has been verified using Mathematica.) Hence, in all three cases

*p*

_{1}(1)<0,

*p*

_{1}(1)=0, and

*p*

_{1}(1)>0, one gets

*σ*(1)=1. For

*x*=−1 we have Here, we obtain

*σ*(−1)=2. Thus, by Sturm’s theorem, the difference

*σ*(−1)−

*σ*(0) gives the number of real zeros of

*Q*in the interval (−1,0] and

*σ*(0)−

*σ*(1) is the number of zeros in (0,1], see Table 3.

The number of real zeros of *Q* in the intervals (−1,0) and (0,1) as it follows from Sturm’s theorem

Range of | | | | (−1,0) | (0,1) | |
---|---|---|---|---|---|---|

| | |||||

0< | 0< | 2 | 2 | 1 | 0 | 1 |

0< | 1/2< | 2 | 1 | 1 | 1 | 0 |

1/2< | 0< | 2 | 1 | 1 | 1 | 0 |

1/2< | 1/2< | 2 | 2 | 1 | 0 | 1 |

*Q*has to every 0<

*τ*<1 precisely one zero in the interval (−1,1) (cf. Table 3 and previous considerations), it follows that to each such zero

*x*=

*x*(

*τ*) there correspond two values of

*α*by means of the trigonometric equation

*Q*(if 0<

*τ*<1), the zero

*x*(

*τ*) is also changing continuously and so are the solutions

*α*

_{1}and

*α*

_{2}. A jump can happen when they are taken modulo 1. We further record that along the vertical lines

*α*=

*u*±1/4 mod 1 one has

*τ*=1/2, and along the vertical lines

*α*=

*u*±1/2 mod 1 and

*α*=

*u*one has

*τ*→0 or

*τ*→1 (if 0<

*v*<1). When identifying the left and right sides of the unit square, these two lines separate the two solutions of (11) in such a way that in each part the points at which

*κ*(

*α*,

*τ*) vanishes form a connected curve varying about the “base lines”

*α*=

*u*±1/4 mod 1. It follows that these curves (together with the boundary of the cylinder) divide the cylinder into two parts in each of which the curvature

*κ*(

*α*,

*τ*) has the same sign. The shapes of these curves do not change when

*is rotated about the polar axis. We may fix*

**w***u*=1/2 and because of the symmetries (including relation

*Q*(

*τ*;

*x*)=−

*Q*(1−

*τ*;−

*x*)) it suffices to consider the curve of the zeros of

*κ*(

*α*,

*τ*) for 0<

*v*<1/2 (recall that these curves are vertical lines for

*v*=1/2) and 0<

*τ*<1/2 which lies in the strip 0<

*α*<1/2. We know that the zero

*x*=

*x*(

*τ*) of

*Q*(we are interested in) in the given setting is in (0,1) (cf. Table 3). Using \(\dot {x}\) to denote the derivative of

*x*with respect to

*τ*, implicit differentiation gives

*Q*′(

*x*(

*τ*))<0, since

*x*(

*τ*) is simple and

*Q*(

*x*) has a negative global minimum for positive

*x*. For

*τ*’s in (0,1/2) at which \(\dot{x}(\tau)\) vanishes, one has

*x*at such

*τ*’s we get The square-bracketed expression is strictly monotonically decreasing on (0,1/2) and evaluates to zero at

*τ*=1/2. Thus, the left-hand side has to be positive for all critical

*τ*in (0,1/2) which in turn implies that \(\ddot{x}(\tau) < 0\) at such

*τ*’s. We conclude that

*x*(

*τ*) has a single maximum in (0,1/2), since it cannot be constant as can be seen from (13) by evaluating

*x*(

*τ*) at that

*τ*′ at which 1−6(1−

*τ*′)

*τ*′=0 and

*x*(

*τ*)→∞ as

*τ*→1/2. By (11) (recall

*u*=1/2)

*α*(

*τ*) has a single maximum in (0,1/2).

## Proposition 19

*The set of zeros of* *κ*(*α*,*τ*) *and the horizontal sides of the unit square divide the unit square either into three parts of equal sign of* *κ*(*α*,*τ*) *or in four parts*, *cf*. *Fig*. 2.

*Q*in (−1,1). The change of variable \(x = 2 \sqrt{-p / 3} \, \cos\theta\) gives the equivalence

*x*(

*τ*)→0 as

*τ*→0 or

*τ*→1. Hence, when moving towards the upper or lower side of the square along the curve of zeros of the curvature, one approaches the corresponding “base line”

*α*=

*u*±1/4 mod 1.

*t*(cf. (8)) we have where

*X*=

*t*−(1−2

*v*)(1−2

*τ*). Reordering the terms and using the substitution

*G*=

*t*−(1−2

*v*), we arrive at The zeros of the numerator of the curvature (10) are determined by a polynomial in

*τ*of degree 4. Thus, there can be at most four pairs (symmetry with respect to

*α*=

*u*) of points on the level set at which the curvature vanishes. For the sake of completeness, in a similar way one obtains

*t*=1−2

*v*, or equivalently

*G*=0), the curvature along the corresponding level curve reduces to

*τ*<1, the corresponding value(s) of

*α*can be obtained from the relation

*τ*is (0,

*τ*

_{1}] with

*τ*

_{1}=4

*v*(1−

*v*). For future reference we record that for 0<

*v*<1 with

*v*≠1/2 the curvature (15) vanishes only for

## Proposition 20

*The curves of zeros of the curvature* (10) (*as functions of* *τ*) *assume their extrema at* *τ* _{ v } *and* 1−*τ* _{ v } *with* *τ* _{ v } *given in* (17).

## Proof

Suppose that *u*=1/2 and 0<*v*<1/2. Then 0<*τ* _{ v }<1/2. On observing that the right-hand side of (16) for *τ*=*τ* _{ v } is also the zero *x*(*τ* _{ v }) in the interval (0,1) of the polynomial *Q*, it can be verified with the help of Mathematica that the right-hand side of (12) vanishes and therefore \(\dot{x}( \tau_{v} ) = 0\); that is, the zero *x*(*τ*) is extremal at *τ*=*τ* _{ v }. Using the symmetry relation *Q*(*τ*;*x*)=−*Q*(1−*τ*;−*x*), the zero *x*(*τ*) is also extremal at *τ*=1−*τ* _{ v }. By means of (16) this translates into extrema of the curve of zeros of the curvature (10). A shift in *u* (rotation of * w* about the polar axis) does not change the shape of the level curves and the general result follows. □

*Q*in (−1,1) can assume, also cf. (13):

*x*(

*τ*

_{ v }) is a strictly monotonically decreasing function in

*v*which is symmetric with respect to

*v*=1/2. Hence \(| x(\tau_{v}) | \leq x(0^{+}) = 1 / \sqrt{2}\). Using this bound in (16) yields that |

*α*−(

*u*±1/4)|≤1/8 (when wrapping around).

*t*=1−2

*v*(associated with the North Pole) has precisely one symmetric (in the cylindrical view) pair of intersection points with the curves of zeros of the curvature function (10) at

*τ*=

*τ*

_{ v }in the strip 0<

*τ*<1. A similar result holds for the level curve associated with the South Pole (

*t*=−(1−2

*v*)).

## Proposition 21

*Let* 0<*v*<1/2. *Then the level curves with* *t* *in the range* −(1−2*v*)≤*t*≤1−2*v* *have precisely one symmetric* (*in the cylindrical view*) *pair of intersection points with the curve of zeros of the curvature function* (10). *For* *t* *in the ranges* −1<*t*<−(1−2*v*) *or* 1−2*v*<*t*<1 *there are either no intersection points*, *one pair of tangential points*, *or two pairs*.

The analogous result holds for 1/2<*t*<1. (For *v*=1/2 the level curves for distance \(\sqrt{2}\) are the verticals at *α*=*u*±1/4mod1 and coincide with the curve of zeros of *κ*(*α*,*τ*) and also coincide with the critical curves. The other level curves have no intersections.)

## Proof

Without loss of generality assume that *u*=1/2. We have already established that either critical level curve has precisely one pair of symmetric (with respect to *α*=*u*) intersection point with the two curves of zero curvature about the base lines *α*=*u*±1/4 at the values *τ*=*τ* _{ v } and *τ*=1−*τ* _{ v }. These parameter values also give the position of the extrema of the zero curves, cf. Fig. 2. The left zero curve \(\mathcal{Z}\) is increasing for *τ* in (0,*τ* _{ v }), decreasing for *τ* in (*τ* _{ v },1−*τ* _{ v }) and increasing again for *τ* in (1−*τ* _{ v },1).

Let −(1−2*v*)<*t*<1−2*v* and *Γ* _{ t } denote the left half of the corresponding level curve starting at the left side at some point (0,*τ* _{1}) and ending at some point (*u*,*τ* _{2}). (The other half is symmetric.) We note that the part where the zero curve is increasing is contained in the regions separated off by the critical level curves. Thus, an intersection between \(\mathcal{Z}\) and *Γ* _{ t } can only occur for *τ* in the interval [*t* _{ v },1−*t* _{ v }]. The curvature along *Γ* _{ t } changes continuously (cf. (14) and subsequent formula) from negative to positive value. Hence, there is an intersection point of \(\mathcal{Z}\) and *Γ* _{ t } and the *Γ* _{ t } cannot change abruptly. In particular, both \(\mathcal{Z}\) and *Γ* _{0} ^{2} pass through (1/4,1/2), which is their only intersection point because for *τ*≠1/2 the vertical line *α*=1/4 separates both curves. A *Γ* _{ t } with 0<*t*<1−2*v* (−(1−2*v*)<*t*<0) has to intersect \(\mathcal{Z}\) in the strip 1/4<*α*<1/2 (0<*α*<1/4). Inspecting the partial derivatives of *F* (cf. (9a)–(9e)) (9c), (9d), (9e)) it follows that the gradient of *F* at the intersection point, which is the outward normal at the level curve *Γ* _{ t }, points into the upper left part; that is, the tangent vector at *Γ* _{ t } at the intersection point shows to the right whereas the tangent vector at \(\mathcal{Z}\) at this point shows to the left. Moreover, if 0<*t*<1−2*v*, then the curve *Γ* _{0} separates \(\mathcal{Z}\) and *Γ* _{ t } for *τ*≥1/2 and in the remaining part both curves \(\mathcal{Z}\) and *Γ* _{ t } bend away from each other because \(\mathcal{Z}\) is decreasing with growing *τ* and the curvature along *Γ* _{ t } becomes positive. Consequently, there is only one intersection point of *Γ* _{ t } and \(\mathcal{Z}\). A similar argument holds for −(1−2*v*)<*t*<0.

Let 1−2*v*<*t*<1. Let *Γ* _{ t } denote the left half of the level curve. For *t* sufficiently close to 1 there is no intersection with \(\mathcal{Z}\) and the level curve is convex. If *Γ* _{ t } and \(\mathcal {Z}\) intersect in only one point, then the level curve is still convex, since the curvature function *κ*(*α*,*τ*) has positive sign in the section between \(\mathcal{Z}\) and the vertical line *α*=*u*. In this case \(\mathcal{Z}\) and *Γ* _{ t } share a common tangent at the intersection point. If the curvature along *Γ* _{ t } changes its sign to negative, then it has to become positive again, since it is positive when crossing the vertical *α*=*u*. But after changing back to positive curvature, both curves are bending away from each other. So, there can be no other intersection point.

By symmetry with respect to the line *α*=*u* one has pairs of symmetric intersection points.

Shifting *u* does not change the form of the curves and their relative positions. This completes the proof. □

## 7 Proofs

## Proof of Lemma 3

For *t*=1 the spherical cap is a point and for *t*=−1 it is the whole sphere. Their pre-images (a point and the whole unit square) are convex. So, we may assume that −1<*t*<1.

*Case (i)*: Let * w* be either the North or the South Pole. Then the pre-images of the boundary of spherical caps centred at

*are horizontal lines in the unit square. Hence, the pre-image of such a spherical cap is convex.*

**w** *Case (ii)*: Let * w* be on the equator (that is,

*v*=1/2). We know that the curvature (10) vanishes along the lines

*α*=

*u*±1/4mod1. First, suppose that

*u*=1/4. Then the pre-image of any spherical cap centred at

*with boundary points at most Euclidean distance \(\sqrt{2}\) away from*

**w***is convex. For a larger spherical cap*

**w***C*it follows that its complement \(\overline{C}\) with respect to the sphere (centred at the antipodal point −

*) has the property that points on the boundary have distance \(\leq\sqrt{2}\) from −*

**w***. Hence, the pre-image of \(\overline{C}\) is convex. When rotating*

**w***about the polar axis (that is, shifting*

**w***u*), the vertical boundaries of the square cut these convex sets into two parts. We conclude that the pre-image of a spherical cap centred at

*or its complement with respect to the sphere is the union of at most two convex sets.*

**w** *Case (iii)*: Let * w* be neither the poles nor located at the equator. Without loss of generality we may assume that

*is in the upper half of the sphere; that is, 0<*

**w***v*<1/2. (Otherwise we can use reflection with respect to the equator.) First, let us consider the canonical position

*u*=1/2. Let −(1−2

*v*)≤

*t*≤1−2

*v*. Then, by Proposition 21, there are precisely two (symmetric) points along the level curve at which the curvature vanishes, say at (

*α*

_{1},

*τ*

_{ t }) and (

*α*

_{2},

*τ*

_{ t }). This yields a decomposition of the unit square into three vertical rectangles such that either the part above or below the level curve is convex. Let 1−2

*v*<

*t*<1. By Proposition 21 the level curve is already convex or there are two pairs of symmetric points at which the curvature along the level curve vanishes and a sign change occurs. Hence, there are numbers

*τ*

_{1}<

*τ*

_{2}such that the level curve is convex for

*τ*≤

*τ*

_{1}and convex for

*τ*≥

*t*

_{2}. The remaining middle part can be covered by a convex isosceles trapezoid which in turn can be split by some vertical line contained in the level set associated with the level curve. Thus, one has again two convex polygons which are divided into a convex and non-convex part by the level curve. A similar argument holds for −1<

*t*<−(1−2

*v*).

*u*does not increase the number of vertical rectangles needed for 0≤

*t*≤1−2

*v*. (In fact, one may even reduce the number of elements of the partition.) In the case 1−2

*v*<

*t*<1 one may need to use a covering of the pre-image of the spherical cap with up to 7 pieces. A more precise analysis is listed in Table 4.

Worst-case admissible convex covering with *p* part and *q* of which are convex. The *vertical lines* show canonical positions of the *vertical borders* of [0,1]^{2}

## Proof of Proposition 8

Radon’s theorem (see e.g. [5, Theorem 4.1]) states that any set of *d*+2 points from ℝ^{ d } can be partitioned into two disjoint subsets whose convex hulls intersect. Particularly, let *A* denote a set of 5 points on the sphere. Then by Radon’s theorem there exists a partitioning of *A* into disjoint subsets whose convex hulls intersect. Thus the set *A* cannot be shattered by the class of half-spaces. Since every spherical cap is the intersection of the sphere with an appropriate half-space, the set *A* can also not be shattered by the class of spherical caps. Thus, the VC dimension of the class of spherical caps is at most 5.

On the other hand, let the set \(\hat{A}\) consist of the points of a regular simplex, which lie on the sphere. Then some simple considerations show that the set \(\hat{A}\) is shattered by the class of spherical caps. Thus, the VC dimension of the class of spherical caps (and therefore of course also the VC dimension of the class \(\mathcal {C}\) of spherical caps for which the centre **w** and the height *t* are rational numbers, which was used in Sect. 4) equals 5. □

## Proof of Theorem 9

*s*≥2

*K*,

## Proof of Theorem 10

*σ*(

*C*

^{∗})=1/2. By the central limit theorem, for any

*t*≥0,

*ε*>0 and sufficiently small

*C*

_{3}(

*ε*)>0,

*N*. Since \(D(Z_{N}) \geq\vert\frac{1}{N} \sum _{n=0}^{N-1} 1_{C^{*}} (X_{n}) - \sigma(C^{*}) \vert\), this implies

*N*.

*ε*>0 and sufficiently large

*C*

_{4}(

*ε*),

*N*. Combining (18) and (19) we obtain

*N*, which proves Theorem 11. □

## Footnotes

## Notes

### Acknowledgements

The authors are grateful to Peter Kritzer and Friedrich Pillichshammer for inspiring suggestions. In particular, F. Pillichshammer pointed out Theorem 14.

The research of C.A. was supported by the Austrian Research Foundation (FWF), Project S9603-N23. The research of J.S.B. was supported by an Australian Research Council Discovery Project. The research of J.D. was supported by an Australian Research Council Queen Elizabeth 2 Fellowship.

## References

- 1.Aistleitner, C.: Covering numbers, dyadic chaining and discrepancy. J. Complex.
**27**(6), 531–540 (2011) MathSciNetMATHCrossRefGoogle Scholar - 2.Alexander, K.S.: The central limit theorem for empirical processes on Vapnik–Červonenkis classes. Ann. Probab.
**15**(1), 178–203 (1987) MathSciNetMATHCrossRefGoogle Scholar - 3.Alexander, K.S., Talagrand, M.: The law of the iterated logarithm for empirical processes on Vapnik–Červonenkis classes. J. Multivar. Anal.
**30**(1), 155–166 (1989) MathSciNetMATHCrossRefGoogle Scholar - 4.Andrievskii, V.V., Blatt, H.-P., Götz, M.: Discrepancy estimates on the sphere. Monatshefte Math.
**128**(3), 179–188 (1999) MATHCrossRefGoogle Scholar - 5.Barvinok, A.: A Course in Convexity. Graduate Studies in Mathematics, vol. 54. American Mathematical Society, Providence (2002) MATHGoogle Scholar
- 6.Beck, J.: Sums of distances between points on a sphere—an application of the theory of irregularities of distribution to discrete geometry. Mathematika
**31**(1), 33–41 (1984) MathSciNetMATHCrossRefGoogle Scholar - 7.Beck, J.: On the discrepancy of convex plane sets. Monatshefte Math.
**105**(2), 91–106 (1988) MATHCrossRefGoogle Scholar - 8.Beck, J., Chen, W.W.L.: Irregularities of Distribution. Cambridge Tracts in Mathematics, vol. 89. Cambridge University Press, Cambridge (2008). Reprint of the 1987 original. [MR0903025] Google Scholar
- 9.Bendito, E., Carmona, A., Encinas, A.M., Gesto, J.M., Gómez, A., Mouriño, C., Sánchez, M.T.: Computational cost of the Fekete problem. I. The forces method on the 2-sphere. J. Comput. Phys.
**228**(9), 3288–3306 (2009) MathSciNetCrossRefGoogle Scholar - 10.Bondarenko, A., Radchenko, D., Viazovska, M.: Optimal asymptotic bounds for spherical designs (2011). arXiv:1009.4407v3 [math.MG]
- 11.Brauchart, J.S.: Optimal logarithmic energy points on the unit sphere. Math. Comput.
**77**(263), 1599–1613 (2008) MathSciNetMATHCrossRefGoogle Scholar - 12.Brauchart, J.S., Dick, J.: Quasi-Monte Carlo rules for numerical integration over the unit sphere \(\mathbb{S}^{2}\). Numer. Math.
**121**(3), 473–502 (2012) MathSciNetMATHCrossRefGoogle Scholar - 13.Brauchart, J.S., Saff, E.B., Sloan, I.H., Womersley, R.S.: Sequences of approximate spherical designs. Manuscript, 62 pages Google Scholar
- 14.Brauchart, J.S., Womersley, R.S.: Numerical integration over the unit sphere, \(\mathbb{L}_{2}\)-discrepancy and sum of distances. Manuscript, 26 pages Google Scholar
- 15.Cassels, J.W.S.: An Introduction to the Geometry of Numbers. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete, vol. 99. Springer, Berlin (1959) MATHCrossRefGoogle Scholar
- 16.Chen, X., Frommer, A., Lang, B.: Computational existence proofs for spherical
*t*-designs. Numer. Math.**117**(2), 289–305 (2011) MathSciNetMATHCrossRefGoogle Scholar - 17.Damelin, S.B., Grabner, P.J.: Energy functionals, numerical integration and asymptotic equidistribution on the sphere. J. Complex.
**19**(3), 231–246 (2003). Numerical integration and its complexity (Oberwolfach, 2001) MathSciNetCrossRefGoogle Scholar - 18.Damelin, S.B., Grabner, P.J.: Corrigendum to: “Energy functionals, numerical integration and asymptotic equidistribution on the sphere” [J. Complexity
**19**(2003), no. 3, 231–246; mr1984111]. J. Complex.**20**(6), 883–884 (2004) MathSciNetGoogle Scholar - 19.Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geom. Dedic.
**6**(3), 363–388 (1977) MathSciNetMATHCrossRefGoogle Scholar - 20.Dick, J., Pillichshammer, F.: Digital Nets and Sequences. Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge University Press, Cambridge (2010) MATHGoogle Scholar
- 21.Dudley, R.M.: Central limit theorems for empirical measures. Ann. Probab.
**6**(6), 899–929 (1979). 1978 MathSciNetCrossRefGoogle Scholar - 22.Durst, M., Dudley, R.M.: Empirical processes, Vapnik–Chervonenkis classes and Poisson processes. Probab. Math. Stat.
**1**(2), 109–115 (1981). 1980 MathSciNetGoogle Scholar - 23.Faure, H.: Discrépance de suites associées à un système de numération (en dimension
*s*). Acta Arith.**41**(4), 337–351 (1982) MathSciNetMATHGoogle Scholar - 24.Götz, M.: On the distribution of weighted extremal points on a surface in
**R**^{d},*d*≥3. Potential Anal.**13**(4), 345–359 (2000) MathSciNetMATHCrossRefGoogle Scholar - 25.Grabner, P.J.: Erdős–Turán type discrepancy bounds. Monatshefte Math.
**111**(2), 127–135 (1991) MathSciNetMATHCrossRefGoogle Scholar - 26.Grabner, P.J., Tichy, R.F.: Spherical designs, discrepancy and numerical integration. Math. Comput.
**60**(201), 327–336 (1993) MathSciNetMATHCrossRefGoogle Scholar - 27.Hannay, J.H., Nye, J.F.: Fibonacci numerical integration on a sphere. J. Phys. A
**37**(48), 11591–11601 (2004) MathSciNetMATHCrossRefGoogle Scholar - 28.Hardin, D.P., Saff, E.B.: Discretizing manifolds via minimum energy points. Not. Am. Math. Soc.
**51**(10), 1186–1194 (2004) MathSciNetMATHGoogle Scholar - 29.Hardin, R.H., Sloane, N.J.A.: McLaren’s improved snub cube and other new spherical designs in three dimensions. Discrete Comput. Geom.
**15**(4), 429–441 (1996) MathSciNetMATHCrossRefGoogle Scholar - 30.Haussler, D.: Sphere packing numbers for subsets of the Boolean
*n*-cube with bounded Vapnik–Chervonenkis dimension. J. Comb. Theory, Ser. A**69**(2), 217–232 (1995) MathSciNetMATHCrossRefGoogle Scholar - 31.Heinrich, S., Novak, E., Wasilkowski, G.W., Woźniakowski, H.: The inverse of the star-discrepancy depends linearly on the dimension. Acta Arith.
**96**(3), 279–302 (2001) MathSciNetMATHCrossRefGoogle Scholar - 32.Korevaar, J.: Fekete extreme points and related problems. In: Approximation Theory and Function Series, Budapest, 1995. Bolyai Soc. Math. Stud., vol. 5, pp. 35–62. János Bolyai Math. Soc., Budapest (1996) Google Scholar
- 33.Korevaar, J., Meyers, J.L.H.: Spherical Faraday cage for the case of equal point charges and Chebyshev-type quadrature on the sphere. Integral Transforms Spec. Funct.
**1**(2), 105–117 (1993) MathSciNetMATHCrossRefGoogle Scholar - 34.Kuipers, L., Niederreiter, H.: Uniform Distribution of Sequences. Pure and Applied Mathematics. Wiley-Interscience, New York (1974) MATHGoogle Scholar
- 35.Larcher, G.: Optimale Koeffizienten bezüglich zusammengesetzter Zahlen. Monatshefte Math.
**100**(2), 127–135 (1985) MathSciNetMATHCrossRefGoogle Scholar - 36.Larcher, G.: On the distribution of
*s*-dimensional Kronecker-sequences. Acta Arith.**51**(4), 335–347 (1988) MathSciNetMATHGoogle Scholar - 37.Larcher, G.: Corrigendum: “On the distribution of
*s*-dimensional Kronecker-sequences” [Acta Arith.**51**(1988), no. 4, 335–347; MR0971085 (90f:11065)]. Acta Arith.**60**(1), 93–95 (1991) MathSciNetMATHGoogle Scholar - 38.Li, X.-J., Vaaler, J.D.: Some trigonometric extremal functions and the Erdős–Turán type inequalities. Indiana Univ. Math. J.
**48**(1), 183–236 (1999) MathSciNetMATHCrossRefGoogle Scholar - 39.Lubotzky, A., Phillips, R., Sarnak, P.: Hecke operators and distributing points on the sphere. I. Commun. Pure Appl. Math.
**39**(S, suppl.), S149–S186 (1986). Frontiers of the Mathematical Sciences: 1985 (New York, 1985) MathSciNetMATHCrossRefGoogle Scholar - 40.Lubotzky, A., Phillips, R., Sarnak, P.: Hecke operators and distributing points on
*S*^{2}. II. Commun. Pure Appl. Math.**40**(4), 401–420 (1987) MathSciNetMATHCrossRefGoogle Scholar - 41.Narcowich, F.J., Sun, X., Ward, J.D., Wu, Z.: LeVeque type inequalities and discrepancy estimates for minimal energy configurations on spheres. J. Approx. Theory
**162**(6), 1256–1278 (2010) MathSciNetMATHCrossRefGoogle Scholar - 42.Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63. SIAM, Philadelphia (1992) MATHCrossRefGoogle Scholar
- 43.Niederreiter, H., Sloan, I.H.: Integration of nonperiodic functions of two variables by Fibonacci lattice rules. J. Comput. Appl. Math.
**51**(1), 57–70 (1994) MathSciNetMATHCrossRefGoogle Scholar - 44.Philipp, W.: Empirical distribution functions and uniform distribution mod 1. In: Proc. Conf. Diophantine Approximation and Its Application, Washington, DC, 1972, pp. 211–234. Academic Press, New York (1973) Google Scholar
- 45.Saff, E.B., Kuijlaars, A.B.J.: Distributing many points on a sphere. Math. Intell.
**19**(1), 5–11 (1997) MathSciNetMATHCrossRefGoogle Scholar - 46.Sjögren, P.: Estimates of mass distributions from their potentials and energies. Ark. Mat.
**10**, 59–77 (1972) MathSciNetMATHCrossRefGoogle Scholar - 47.Sloan, I.H., Womersley, R.S.: Extremal systems of points and numerical integration on the sphere. Adv. Comput. Math.
**21**(1–2), 107–125 (2004) MathSciNetMATHCrossRefGoogle Scholar - 48.Sloan, I.H., Womersley, R.S.: A variational characterisation of spherical designs. J. Approx. Theory
**159**(2), 308–318 (2009) MathSciNetMATHCrossRefGoogle Scholar - 49.Sobol’, I.M.: Distribution of points in a cube and approximate evaluation of integrals. Z̆. Vyčisl. Mat. i Mat. Fiz.
**7**, 784–802 (1967) MathSciNetGoogle Scholar - 50.Stolarsky, K.B.: Sums of distances between points on a sphere. II. Proc. Am. Math. Soc.
**41**, 575–582 (1973) MathSciNetMATHCrossRefGoogle Scholar - 51.Talagrand, M.: Sharper bounds for Gaussian and empirical processes. Ann. Probab.
**22**(1), 28–76 (1994) MathSciNetMATHCrossRefGoogle Scholar - 52.Talagrand, M.: Vapnik–Chervonenkis type conditions and uniform Donsker classes of functions. Ann. Probab.
**31**(3), 1565–1582 (2003) MathSciNetMATHCrossRefGoogle Scholar - 53.Vapnik, V.N., Červonenkis, A.J.: The uniform convergence of frequencies of the appearance of events to their probabilities. Teor. Verojatnost. i Primenen.
**16**, 264–279 (1971) MathSciNetMATHGoogle Scholar - 54.Wagner, G.: Erdős–Turán inequalities for distance functions on spheres. Mich. Math. J.
**39**(1), 17–34 (1992) MATHCrossRefGoogle Scholar