Point distribution in compact metric spaces, III. Two-point homogeneous spaces

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by  M. M. Skriganov

Point distribution in compact metric spaces,
III. Two-point homogeneous spaces
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M. M. Skriganov

n
a
St.Petersburg Department of
J
Steklov Mathematical Institute
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1 Russian Academy of Sciences

] E-mail: [email protected]
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t
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1 We continue the investigation of point distributions in compact metric
v
spacesstartedinthepapers[28,29]. Ourconcerniswithdiscrepancies ofsuch
5
4 distributions in metric balls and sums of pairwise distances between points of
5 distributions. In the present paper we consider compact connected two-point
4
0 homogeneous spaces (Riemannian symmetric spaces of rank one). All such
. spaces are known, they are spheres in the Euclidean spaces, the real, complex
1
0 and quaternionic projective spaces and the octonionic projective plane.
7
Using the geometric features of two-point spaces, we show that Sto-
1
: larsky’s invariance principle, well-known for the Euclidean spheres, can be
v
i extended to all projective spaces and the octonionic projective plane. Rely-
X
ing on the theory of spherical functions, we obtain the best possible bounds
r
a for quadratic discrepancies and sums of distances for point distributions in
the two-point homogeneous spaces. Applications to t-designs and L´evy-
Schoenberg kernels in such spaces are also considered in the paper.
Key words: geometry of distances, uniform distributions, t-designs ,
two-point homogeneous spaces
Contents
A. Main results
1. Discrepancies and metrics
2. The strong invariance principle and bounds for
discrepancies and sums of distances
3. Applications to t-designs
4. Remarks on the L´evy-Schoenberg kernels
B. Geometry of two-point homogeneous spaces
and the strong invariance principle
5. Models of projective spaces and chordal metrics
6. Proof of Theorem 2.1
7. Proof of Lemma 2.1
C. Spherical functions and bounds for discrepancies
and sums of distances
8. Commutative spaces and spherical functions
9. Spherical function expansions for discrepancies and metrics
10. Bounds for Fourier-Jacobi coefficients
11. Proof of Theorems 2.2 and 3.1
References
A. Main results
1. Discrepancies and metrics
Let be a compact connected metric space with a fixed metric θ and a
M
finite Borel measure µ, normalized by
diam( ,θ) = π, µ( ) = 1, (1.1)
M M
where
diam( ,ρ) = sup ρ(x ,x ) : x ,x (1.2)
1 2 1 2
E { ∈ E}
denotes the diameter of a subset with respect to a metric ρ.
E ⊆ M
We write B (y) = x : θ(x,y) < r for the ball of radius r [0,π]
r
{ } ∈
centered at y and of volume v (y) = µ(B (y)). Since the space is
r r
∈ M M
connected, we have = [0,π], where = r = ρ(x ,x ) : x ,x is
1 2 1 2
R R { ∈ M}
the set of all possible radii.
2
Let be a finite subset consisting of N points (not necessary
N
D ⊂ M
different). The local discrepancy of the subset in the ball B (y) is defined
N r
D
by
Λ[B (y), ] = # B (y) Nv (y)
r N r N r
D { ∩D }−
= Λ(B (y),x), (1.3)
r
xX∈DN
where
Λ(B (y),x) = χ(B (y),x) v (y), (1.4)
r r r
−
and χ( ,x) denotes the characteristic function of s subset .
E E ⊂ M
The quadratic discrepancies are defined by
λ [ ] = Λ[B (y), ]2dµ(y) = λ (x ,x ), (1.5)
r N r N r 1 2
D D
MZ x1,Xx2∈DN
where
λ (x ,x ) = Λ(B (y),x )Λ(B (y),x )dµ(y), (1.6)
r 1 2 r 1 r 2
Z
M
and
π
λ[η, ] = λ [ ]η(r)dr = λ(η,x ,x ), (1.7)
N r N 1 2
D D
Z0 x1,Xx2∈DN
where
π
λ(η,x ,x ) = λ (x ,x )η(r)dr, (1.8)
1 2 r 1 2
Z
0
where η(r), r [0,π], is a non-negative weight function. The quantities
∈
λ [ ]1/2 and λ[η, ]1/2 are known as L -discrepancies. In the present
r N N 2
D D
paper it is more convenient to deal with the quadratic discrepancies (1.5)
and (1.7).
We introduce the following extremal characteristic
λ (η) = infλ[η, ], (1.9)
N N
D D
N
where the infimum is taken over all N-point subsets .
N
D ⊂ M
3
For any metric ρ on we put
M
ρ[ ] = ρ(x ,x ), (1.10)
N 1 2
D
x1,Xx2∈DN
and introduce yet another extremal characteristic
ρ = supρ[ ], (1.11)
N N
D
D
N
where the supremum is taken over all N-point subsets .
N
D ⊂ M
We write ρ for the average value of a metric ρ,
h i
ρ = ρ(y ,y )dµ(y )dµ(y ). (1.12)
1 2 1 2
h i
ZZ
M×M
Introduce the following symmetric difference metrics on the space
M
π
θ∆(η,y ,y ) = θ∆(y ,y )η(r)dr, (1.13)
1 2 r 1 2
Z
0
where
1
θ∆(y ,y ) = µ(B (y )∆B (y ))
r 1 2 2 r 1 r 2
1
= v (y )+v (y ) 2µ(B (y ) B (y )) . (1.14)
r 1 r 2 r 1 r 2
2 − ∩
(cid:16) (cid:17)
Here
B (y )∆B (y ) = B (y ) B (y ) B (y ) B (y ) (1.15)
r 1 r 2 r 1 r 2 2 1 r 2
∪ \ ∩
is the symmetric difference of the balls B (y ) and B (y ).
r 1 r 2
The symmetry of the metric θ implies the following useful relation
χ(B (y),x) = χ(B (x),y) = χ(r θ(x,y)) = χ (θ(x,y)) (1.16)
r r r
−
where χ(z), z R is the characteristic function of the half-axis [0, ), and
∈ ∞
χ ( ) is the characteristic function of the interval [0,r], 0 r π. Using
r
· ≤ ≤
4
(1.14), (1.15), (1.16), we can write
1
θ∆(y ,y ) = χ(B (y )∆B (y ))dµ(y)
r 1 2 2 r 1 r 2
Z
M
1
= [χ(B (y ),y)+χ(B (y ),y) 2χ(B (y ),y)χ(B (y ),y)]dµ(y)
r 1 r 2 r 1 r 2
2 −
Z
M
1
= χ(B (y ),y) χ(B (y ),y) dµ(y) (1.17)
r 1 r 2
2 | − |
Z
M
For the average values (1.12) of metrics (1.14) and (1.13) we obtain
π
θ∆(η) = θ∆ η(r)dr, (1.18)
h i h r i
Z
0
1
θ∆ = θ∆(y ,y )dµ(y )dµ(y ) = (v (y) v (y)2)dµ(y) (1.19)
h r i 2 r 1 2 1 2 r − r
ZZ Z
M×M M
Thesymmetric difference ofanytwo subsets coincides withthesymmetric
difference of their complements. Hence
1
θ∆(y ,y ) = µ(B′(y )∆B′(y ))
r 1 2 2 r 1 r 2
1
= v′(y )+v′(y ) 2µ(B′(y ) B′(y )) , (1.20)
2 r 1 r 2 − r 1 ∩ r 2
(cid:16) (cid:17)
where B′(y) = B (y) is the complement of the ball B (y), and
r M\ r r
v′(y) = µ(B′(y)) = 1 v (y). (1.21)
r r − r
Now the relation (1.19) takes the form
θ∆ = v (y)v′(y)dµ(y) (1.22)
h r i r r
Z
M
In (1.17) the balls B (y) can be also replaced by their complements B′(y).
r r
The study of the characteristics (1.9) and (1.11) falls within the subjects
of the discrepancy theory and geometry of distances. An extensive literature
5
is devoted to such studies of point distributions on spheres in the Euclidean
space, see, forexamples, [1,2,5,6,12,27]. Itwasshown inourrecent paper[28]
that nontrivial results on the quantities (1.9) and (1.11) can be obtained for
very general metric spaces. Some of these results, needed for the present
work, are given below in Theorems 1.1 and 1.2.
A metric space is called distance-invariant, if the volume of any ball
M
v = v (y) is independent of y , see [23]. For such spaces the formulas
r r
∈ M
for the discrepancies (1.5) and the symmetric difference metrics (1.14) can
be essentially simplified. Substituting (1.14) into (1.6), we obtain
λ (y ,y ) = χ(B (y ),y)χ(B (y ),y)dµ(y) v2
r 1 2 r 1 r 2 − r
Z
M
= µ(B (y ) B (y ) v2, (1.23)
r 1 ∩ r 2 − r
and correspondingly,
λ [ ] = µ(B (y ) B (y )) v2N2. (1.24)
r DN r 1 ∩ r 2 − r
y1,Xy2∈DN
Similarly, the relations (1.14), (1.20) and (1.19), (1.22) take the form
θ∆(y ,y ) = v χ(B (y ),y)χ(B (y ),y)dµ(y)
r 1 2 r − r 1 r 2
Z
M
= v µ(B (y ) B (y )) = v′ µ(B′(y ) B′(y )), (1.25)
r − r 1 ∩ r 2 r − r 1 ∩ r 2
θ∆ = v v2 = v v′, (1.26)
h r i r − r r r
and
θ∆[ ] = v N2 µ(B (y ) B (y )). (1.27)
r DN r − r 1 ∩ r 2
y1,Xy2∈DN
Integrating these relations with η(r), r [0,π], one can obtain the corre-
∈
sponding formulas for the quantities (1.7), (1.8), (1.13), (1.18).
Thetypical examplesofdistance-invariant spacesarehomogeneousspaces
= G/K, where Gisacompact group,K Gisaclosed subgroup, whileθ
M ⊂
and µ are G-invariant metric and measure on . In this case, the quantities
M
6
(1.6), (1.8) and (1.13), (1.14) are also G-invariant:
λ (gy ,gy )= λ (y ,y ), λ(η,gy ,gy ) = λ(η,y ,y ),
r 1 2 r 1 2 1 2 1 2
θ∆(gy ,gy )=θ∆(gy ,gy ),θ∆(η,gy ,gy )=θ∆(η,y ,y ),
r 1 2 r 1 2 1 2 1 2 
µ(B (gy ) B (gy ))=µ(B (y ) B (y )),y ,y µ,g G. 
r 1 r 2 r 1 r 2 1 2
∩ ∩ ∈ ∈
(1.28)


Comparing the relations (1.23)–(1.27), we arrive to the following result.
This result and its generalizations were given in [29, Thms. 2.1, 3.1].
Theorem 1.1. ( The weak invariance principles). Let a compact connected
metric space with a metric θ and a measure µ be distance-invariant. Then
M
we have
λ (y ,y )+θ∆(y ,y ) = θ∆ , (1.29)
r 1 2 r 1 2 h r i
λ(η,y ,y )+θ∆(η,y ,y ) = θ∆(η) , (1.30)
1 2 1 2
h i
λ(η, )+θ∆(η, ) = θ∆(η) N2, (1.31)
N N
D D h i
λ (η)+θ∆(η) = θ∆(η) N2. (1.32)
N N h i
Here r = [0,π] and is an arbitrary N-point subset. The
N
∈ R D ⊂ M
equalities (1.30), (1.31) and (1.32) hold with any weight function η such that
the integrals (1.7), (1.8) and (1.13), (1.18) converge.
Obviously, the integrals (1.7), (1.8) and (1.13), (1.18) converge for any
wight function η summable on the interval [0,π]. More general conditions of
convergence of these integrals for two-point homogeneous spaces are given in
Lemma 2.1(i) below.
The strong invariance principle for two-point homogeneous spaces will be
established in the next section in Theorem 2.2. Our terminology of strong
and weak invariance principles is explained in comments to Theorem 2.2.
A compact metric space with a metric θ and a measure µ is called
M
d-rectifiable if there exist a measure ν on the d-dimensional unit cube
Id = [0,1]d absolutely continuous with respect to thed-dimensional Lebesgue
measure on Id, a measurable subset Ø Id, and an injective Lipschitz map-
⊂
ping f : Ø , such that µ( f(Ø)) = 0; and µ( ) = ν(f−1( f(Ø))
→ M M\ E E ∩
for any µ-measurable subset . Recall that a map f : Ø Rd is
E ⊂ M ⊂ → M
Lipschitz if
θ(f(Z ),f(Z )) c Z Z , Z ,Z Ø, (1.33)
1 2 1 2 1 2
≤ k − k ∈
7
with a positive constant c, and the smallest such constant is called the Lips-
chitzconstantoff anddenotedbyLip(f); in(1.33) denotestheEuclidean
k·k
metric in Rd, cf [26].
Notice that any smooth (or piece-wise smooth) compact d-dimensional
manifold is d-rectifiable if in the local coordinates the metric satisfies (1.12),
and the measure is absolutely continuous with respect to the d-dimensional
Lebesgue measure. Particularly, any compact d-dimensional Riemannian
manifold with the geodesic metric θ and the Riemannian measure µ is d-
rectifiable. In this case, it is known that condition (1.12) holds; see [21,
Chapter I, Proposition 9.10]. On the other hand, the condition on the Rie-
mannian measure is obvious because the metric tensor is continuous.
The following result was established in [29, Theorem.4.2]. Notice that
the proof of this result is relying on a probabilistic version of Theorem 1.1,
see [29, Theorem 3.1].
Theorem 1.2. Suppose that a compact metric space , with a metric θ
M
and a measure µ, is d-rectifiable. Write C = d2d−1Lip(f), where Lip(f) is
the Lipschitz constant of the map f in the definition of d-rectifiability of the
space . Then the following hold:
M
(i) If a metric ρ on satisfies the inequality
M
ρ(x ,x ) c θ(x ,x ) (1.34)
1 2 0 1 2
≤
with a constant c > 0, then
0
ρ ρ N2 c CN1−1/d. (1.35)
N 0
≥ h i −
(ii) If the metric θ∆(η) satisfies the inequality
θ∆(η,x ,x ) c θ(x ,x ) (1.36)
1 2 0 1 2
≤
with a constant c > 0, then
0
θ∆(η) θ∆(η) N2 c CN1−1/d (1.37)
N ≥ h i − 0
and
λ (η) c CN1−1/d. (1.38)
N 0
≤
Undersuch generalassumptionsonecannotexpect thatthebounds(1.37)
and (1.38) are best possible. The corresponding counterexample is given
8
in the next section, see the relation (2.22). Theorem 1.2 guarantees the
existence of well-distributed point subsets in all compact d-rectifiable spaces.
It should be emphasize that a very non-trivial and difficult problem is to
construct such uniformly distributed subsets explicitly. For spheres in the
Euclidean space a deep investigation of this problem has been given in [25].
In the present paper we will show that the bounds (1.37) and (1.38) are
best possible for compact connected two-point spaces and general classes of
weight functions η, see Theorem 2.2 below. Main results of this paper were
stated previously in [29].
2. The strong invariance principle and bounds
for discrepancies and sums of distances
In this section we state and discuss our main results on strong invariance
principles and best possible bounds for discrepancies and sums of distances
on two-point homogeneous spaces.
Recall some necessary facts on two-point homogeneous spaces, see [7,21,
22,33,34]. Additional facts on the geometry and harmonic analysis on such
spaces will be given in sections 5 and 8.
Let G = G( ) be the group of isometries of a metric space with a
M M
metric θ, i.e. θ(gx ,gx ) = θ(x ,x ) for all x , x and g G. The
1 2 1 2 1 2
∈ M ∈
space is called two-point homogeneous, if for any two pairs of points x ,
1
M
x and y , y with θ(x ,x ) = θ(y ,y ) there exists an isometry g G,
2 1 2 1 2 1 2
∈
such that y = gx , y = gx . In this case, the group G is transitive on
1 1 2 2
and = G/K is a homogeneous space, where the subgroup K G is
M M ⊂
the stabilizer of a point x . Furthermore, the homogeneous space
0
∈ M M
is symmetric, i.e. for any two points y , y there exists an isometry
1 2
∈ M
g G, such that gy = y , gy = y .
1 2 2 1
∈
We consider compact connected two-point homogeneous spaces =
M
G/K. For such spaces G and K G are Lie groups and = G/K are
⊂ M
Riemannian symmetric spaces of rank one. All such spaces are classified
completely, see [33, Sec. 8.12]. They are the following:
(i) The d-dimensional spheres in the Euclidean space Sd Rd+1, Sd =
⊂
SO(d+1)/SO(d) 1 , d 2, and S1 = O(2)/O(1) 1 .
×{ } ≥ ×{ }
(ii) The real projective spaces RPn = O(n+1)/O(n) O(1).
×
(iii) The complex projective spaces CPn = U(n+1)/U(n) U(1).
×
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(iv) The quaternionic projective spaces HPn = Sp(n+1)/SP(n) Sp(1),
×
(v) The octonionic projective plane OP2 = F /Spin(9).
4
Here we use the standard notation from the theory of Lie groups; par-
ticularly, F is one of the exceptional Lie groups in Cartan’s classification.
4
see [21,22,33,34].
The indicated projective spaces FPn as compact Riemannian manifolds
have dimensions d,
d = dim FPn = nd , d = dim F, (2.1)
R 0 0 R
where d = 1,2,4,8 for F = R, C, H, O, correspondingly.
0
For spheres Sd we put d = d by definition. Projective spaces of di-
0
mension d ( n = 1) are isomorphic to the spheres Sd0: RP1 S1,CP1
0
≈ ≈
S2,HP1 S4,OP1 S8. We can conveniently agree that d > d (n 2) for
0
≈ ≈ ≥
projective spaces, while the equality d = d holds only for spheres. Under
0
this convention, the dimensions d = nd and d define uniquely (up to iso-
0 0
morphism) the corresponding two-point homogeneous space which we denote
by Q = Q(d,d ). We write θ for the geodesic distance and µ for the Rie-
0
mannian measure on Q(d,d ). The metric θ and the measure µ are invariant
0
under the action of the corresponding group of isometries and normalized by
(1.1). In what follows we always assume that n = 2 if F = O. Projective
spaces OPn do not exist for n > 2. In more detail the geometry of spaces
FPn will be outlined in section 5.
Any space Q(d,d ) is distance-invariant and the volume of balls is given
0
by
r
1 1
v = κ(d,d ) (sin u)d−1(cos u)d0−1du, r [0,π],
r 0
2 2 ∈ 
Z (2.2)
0 


Γ(d/2+d /2) 
κ(d,d ) = B(d/2,d /2)−1 = 0 . 
0 0
Γ(d/2)Γ(d /2)
0



Here B( , ) and Γ( ) are beta and gamma functions, and wehave v =
 π
· · ·
µ(Q(d,d )) = 1. Notice that the different equivalent forms of the relation
0
(2.2) can be found in the literature, see [22, pp. 165–168], [19, pp. 177–
178], [23, pp. 508–510].
From the formula (2.2) we obtain the following two-side bounds
v rd, v′ = 1 v (π r)d0, r [0,π]. (2.3)
r ≃ r − r ≃ − ∈
10