Pair correlation and equidistribution on manifolds

This note is motivated by a series of recent papers that show that if a given deterministic sequence on a circle has a Poisson pair correlation measure, then the sequence is uniformly distributed. Analogous results have been proved for point sequences on higher-dimensional tori. The purpose of this paper is to describe a simple statistical argument that explains this observation and furthermore permits a generalisation to bounded Euclidean domains as well as compact Riemannian manifolds.


BOUNDED DOMAINS
Let Ω ⊂ R d be bounded with vol ∂Ω = 0, where vol denotes the Lebesgue measure in R d . (All subsets of R d in this paper are assumed to be Borel sets.) Consider the triangular array ξ = (ξ i j ) i j with coefficients ξ i j ∈ Ω and indices i, j ∈ N, j ≤ N i , for some given N i ∈ N such that N i < N i+1 .
Example 1. Let Ω = [0, 1]. Take a real sequence (a j ) j∈N and set ξ i j = 〈a j 〉 for j ≤ N i = i, where 〈x〉 denotes the fractional part of x.
Example 2. Let Ω = B d 1 be the open unit ball centered at the origin. Take a sequence (a j ) j∈N in R d such that a j → ∞, and set ξ i j = T −1 i a j , with N i = #{ j | a j < T i } and T 1 < T 2 < . . . → ∞ increasing sufficiently fast so that N j+1 > N j .
We associate with the ith row of ξ the Borel probability measure ν i on Ω, defined by (1) where f ∈ C b (Ω) (bounded and continuous). In other words, ν i represents N i normalised point masses at the points ξ i1 , . . ., ξ iN i . Given a Borel probability measure σ on Ω, we say the triangular array ξ is equidistributed in (Ω, σ) if ν i converges weakly to σ; that is, (2) lim i→∞ ν i f = σ f for every f ∈ C b (Ω).
In the case of Example 1, equidistribution in ([0, 1), vol) corresponds to the classical notion of uniform distribution mod 1.
Let A : clΩ → GL(d, R) be a continuous map. This means in particular that ∆(x) = | det A(x)| is bounded above and below by positive constants. Define the finite Borel measure σ on Ω by (3) σ(d x) = ∆(x) dx.
By multiplying A with a suitable scalar constant, we may assume without loss of generality that σ(Ω) = 1. The role of A in this paper is to set a local frame, at each point x ∈ Ω, relative to which we measure correlations in the array ξ. This is particularly relevant in Section 2, where we extend the present discussion to manifolds. The simplest example of A to keep in mind for now is the constant function A(x) = vol(Ω) −1/d I d (I d is the identity matrix), so that σ(d x) = vol(Ω) −1 dx is the uniform probability measure on Ω.
Given an increasing sequence M = (M i ) i in R >0 , the pair correlation measure ρ i of ξ is defined by where f ∈ C + c (R d ) (non-negative, continuous with compact support). The sequence M determines the scale on which we measure correlations, and A(ξ i j 1 ) provides a local rescaling of length units near each point ξ i j 1 , relative to the density of the measure σ. We call the pair (A, M) a scaling.
If equidistribution (2) is known for some probability measure σ with continuous density ∆, then the most canonical choice for A is A(x) = ∆ 1/d (x)I d and M i = N i , so that (4) captures correlations in units of the average Euclidean distance between the ξ i j near x, which is proportional to (N i ∆(x)) −1/d . The point of the present discussion is, however, that we do not assume equidistribution of the array ξ, and hence there is no a priori preferred choice of A or σ.
Note that ρ i is a locally finite Borel measure on R d . It is not a probability measure. We equip the space of locally finite Borel measures on R d with the vague topology, and say ξ has limiting pair correlation measure ρ for the scaling (A, M), if ρ i converges vaguely to ρ. That is, if We say ρ i has a Poisson limit for the scaling (A, M) if (5) holds with ρ = vol. (The constant multiplier in this relation seems arbitrary, but is in fact determined by our scaling of A such that σ(Ω) = 1.) In this case (5) is equivalent to the statement and Z 2 = = Z 2 \ {( j, j) | j ∈ Z}. We furthermore say ρ i has a sub-Poisson limit if which again is equivalent to the corresponding statement for bounded D ⊂ R d with vol ∂D = 0. In many applications one considers only the pair correlation with respect to the distance between points. We consider here dist(x, y) = x − y , with · the Euclidean norm in R d . The corresponding pair correlation is a locally finite Borel measure on R ≥0 defined by for h ∈ C + c (R ≥0 ). In the spatial statistics literature variants of this are often referred to as Ripley's K -function; cf. [7,Sect. 8.3].
Define the Borel measure ω on R ≥0 by where B d 1 is the open unit ball. We say ρ i has a Poisson limit if it converges vaguely to ω. Note Thus if ρ i has a Poisson limit in the vague topology, then so does ρ i . We say ρ i has a sub-Poisson limit if The latter statement is equivalent to for every r > 0. Theorem 1. Fix A and σ as defined above, and let ξ be a triangular array in Ω. Then the following holds.
(i) Suppose there is a sequence M with M i → ∞ and M i ≤ N i , such that ρ i has a sub-Poisson limit for the scaling (A, M). Then ξ is equidistributed in (Ω, σ).
such that ρ i has a Poisson limit for the scaling (A, M).
It is well known that equidistribution does not imply a Poisson pair correlation at the scale M i = N i . (An elementary example is the triangular array in [0, 1] given by ξ i j = j N i .) Furthermore, a Poisson pair correlation at this scale does not imply that other fine-scale statistics, such as the nearest-neighbour distribution, are Poisson [2,3,4].
The proof of part (i) is split into four lemmas. For x ∈ R d , define the counting measureμ x i on R d by is the open ball of radius ǫ centered at the origin. The Tietze extension theorem allows us to extend ∆ to a continuous function R d → R >0 . We also extend σ to a locally finite measure outside Ω via relation (3).
It is convenient to work with the following normalised variant ofμ x i , Lemma 1. Fix a triangular array ξ, a sequence M with M i → ∞ and M i ≤ N i , and a bounded set D ⊂ R d . Then, for ǫ > 0, Proof. Since ∆ is uniformly continuous, we have For M i sufficiently large, we have This implies (16). Lemma 2. Fix a triangular array ξ, a sequence M with M i → ∞ and M i ≤ N i , and a bounded set D ⊂ R d with vol D > 0. If for every Borel probability measure λ on Ω with density in C c (Ω) we have then ξ is equidistributed.
Proof. Let f ∈ C c (Ω) be the density of λ with respect to σ. Then (19) states explicitly that which by linearity in fact holds for any f ∈ C c (Ω), not necessarily probability densities. Since f and A are uniformly continuous and D is bounded, we have uniformly for y ∈ Ω, as i → ∞. Therefore, .
This relation can be extended to f ∈ C b (Ω) by noting that (23) holds trivially for every constant test function: Any f ∈ C b (Ω) can be approximated from below by a function in C c (Ω), and from above by a function in C c (Ω) plus a constant. This proves that ξ is equidistributed.

Lemma 3. Fix a triangular array ξ and a bounded set D
for every Borel probability measure λ with square-integrable density (with respect to σ).
Proof. Let f be the density of λ. By the Cauchy-Schwarz inequality, . This converges to zero as i → ∞, which proves (25).
Then f ∈ C + c (R d ) and we have, for ǫ > 0, Proof. By Lemma 1, and so Furthermore, by the same reasoning as in the proof of Lemma 1, The summation over distinct indices j 1 = j 2 yields ρ i f with f as defined in (27). The summation over j 1 = j 2 yields M i N i vol D. The function f is compactly supported, since D is bounded. To prove continuity, note that for x − y < ǫ, | f (x) − f (y)| is bounded above by the volume of the ǫ-neighbourhood of ∂D. Continuity of f is therefore implied by the assumption vol ∂D = 0.
Proof of Theorem 1 (i). Assume that ρ i has a sub-Poisson limit for some sequence M with M i ≤ N i . It follows from (12) that, for any δ > 0, ρ i also has a sub-Poisson limit for the scaling (A, By assumption ρ δ i has a sub-Poisson limit. Hence and so With this, Lemma 4 shows that, for any ǫ, δ > 0, Since vol ∂Ω = 0 and thus σ(∂Ω) = 0, we have σ(Ω ǫ ) → σ(Ω) = 1 as ǫ → 0. Thus there is a sequence of δ i → 0, such that for the scaling (A, M ′ ) given by M i This confirms the hypothesis of Lemma 3 for the sequence M ′ . Lemma 3 in turn establishes the assumption for Lemma 2, which completes the proof of claim (i).
Proof of Theorem 1 (ii). Since ξ is equidistributed in (Ω, σ) we have, for ψ ∈ C b (Ω × Ω), Since ψ is bounded, the above statement remains valid with the diagonal terms j 1 = j 2 removed. For fixed M 0 > 0 and f ∈ C + c (R d ), apply this asymptotics with the choice ψ(x 1 , , which is bounded continuous. This yields, The right hand side can be written as Since f , A are continuous and Ω has boundary of Lebesgue measure zero, this expression converges, as M 0 → ∞, to (39) This proves that there is a slowly growing sequence M i → ∞ such that which proves part (ii) of the theorem.

RIEMANNIAN MANIFOLDS
Let (M , g) be a compact Riemannian manifold with metric g. We denote by vol g the corresponding Riemannian volume, and normalise g such that vol g M = 1. The geodesic distance between x, y ∈ M is denoted dist g (x, y). Now consider a triangular array ξ with coefficients in M , and define the corresponding pair correlation measure by In other words, for r > 0, We say ρ (g) i has a Poisson limit for the scaling M if it converges vaguely to ω, with ω as defined in (10) (with vol still the Lebesgue measure in R d ), and similarly say it has a sub-Poisson limit if for which is equivalent to the statement for every r > 0.
The following is a corollary of Theorem 1. Part (i) is closely related to, but not implied by, the results in [8] for the choice ρ Proof of (i). Consider an atlas {(U α , ϕ α ) | α ∈ A } with A finite. We take ϕ α (U α ) ⊂ R d to lie in the same copy of R d , arranged in such a way that the ϕ α (U α ) are pairwise disjoint. Now consider a partition of M by the bounded sets V β with β ∈ B and B finite, so that We assume the partition is sufficiently refined so that for β ∈ B there is a choice of α(β) ∈ A such that cl V β ⊂ U α(β) . We set Ω β = ϕ α(β) V β . The disjoint union is a bounded subset of R d with vol ∂Ω = 0. Given a triangular array ξ in M we define a corresponding array ξ ′ , whose ith row (ξ ′ i j ) j≤N i is given by the elements in the set By Gram-Schmidt orthonormalisation, there is a continuous function A : cl Ω → GL(d, R) such that the metric g is given at x ∈ U α by the positive definite bilinear form where 〈 · , · 〉 is the standard Euclidean inner product. With this choice, and the probability measure σ defined as in (3), we see that the triangular array ξ is equidistributed in (M , vol g ) if and only if ξ ′ is equidistributed in (Ω, σ).
Let us now compare the pair correlation measure ρ i for ξ ′ in Ω as defined in (9) ∈ Ω β ′ with β = β ′ and M i is sufficiently large. This means that the pairs ( j 1 , j 2 ) contributing to ρ i form a subset of those contributing to ρ for ϕ α x − ϕ α y → 0. Both facts taken together imply, by the uniform continuity of h ∈ C + c (R ≥0 ), that This shows that if ρ i has a sub-Poisson limit then so does ρ i . Theorem 1 tells us that therefore ξ ′ is equidistributed in (Ω, σ), and hence (as noted earlier) ξ is equidistributed in (M , vol g ). This yields claim (i).
The limit M 0 → 0 can be calculated in local charts, which leads to the same calculation as in the proof of Theorem 1 (ii).

APPENDIX A. FLAT TORI
It is instructive to adapt the discussion in Section 1 to the case of a multidimensional torus T. This provides an alternative approach to the results in [10]. We represent the torus as T = R d /L , with L ⊂ R d a Euclidean lattice of unit covolume (for example the integer lattice L = Z d ). The required modifications are as follows.
A. Replace Ω by T throughout Section 1, and note that C(T) = C b (T) = C c (T). B. The coefficients of the triangular array are written as ξ i j + L ∈ T with ξ i j ∈ R d . C. Set for simplicity A(x) = I d , so that σ = vol is the uniform probability measure on T. (It is of course possible to adapt the argument also for general continuous A : T → GL(d, R).)