CRYSTALLINE MEASURES IN TWO DIMENSIONS

: Some crystalline measures supported by a Delone set Λ ⊂ R 2 are constructed in this note. This gives a new proof of a remarkable theorem by Pavel Kurasov and Peter Sarnak. 2020 Mathematics Subject Classiﬁcation: Primary: 42A32; Secondary: 43A46.


Introduction
An atomic measure µ on R n is a crystalline measure if the following three conditions are satisfied: (i) µ is supported by a locally finite set, (ii) µ is a tempered distribution, and (iii) the distributional Fourier transform µ of µ is also an atomic measure supported by a locally finite set. This is equivalent to a generalized Poisson summation formula, as is shown in [3]. In the late 1950s crystalline measures were defined under other names and studied by Andrew P. Guinand [1] and Jean-Pierre Kahane and Szolem Mandelbrojt [3]. These authors were motivated by the relationship between (a) the functional equation satisfied by the Riemann zeta function and (b) the standard Poisson formula. More generally let µ = λ∈Λ c(λ)δ λ be a crystalline measure on R n supported by a locally finite set Λ and let µ = s∈S a(s)δ s . We consider the Dirichlet series ζ(µ, s) = {λ∈Λ, λ =0} c(λ)|λ| −s in the complex variable s. Let γ n = π −n/2 2 1−n Γ(n/2) , where Γ denotes the Euler Gamma function. Then ζ(µ, s) is an entire function in the complex plane if 0 / ∈ S, while ζ(µ, s) − γ n a(0) s−n can be extended as an entire function of s ∈ C if 0 ∈ S [3], [7]. The connection between the Poisson summation formula and the properties of the Riemann zeta function is developed in Titchmarsh's treatise and can be traced back to Riemann. When Kahane and Mandelbrojt wrote their paper it was debated whether or not a crystalline measure is necessarily a generalized Dirac comb. But this same year Guinand discovered a revolutionary crystalline measure. In his seminal work [1] Guinand described an explicit atomic measure µ supported by the set Λ = {± k + 1/9, k ∈ N ∪ {0}} and claimed that µ = µ. This beautiful example of a crystalline measure is rooted in Guinand's work on number theory. The fascinating problems raised by Guinand were forgotten for more than fifty years. Fortunately in 2015 Nir Lev and Alexander Olevskii gave a new life to Guinand's work and constructed a crystalline measure which is not a generalized Dirac comb [5]. This was important since the proof given by Guinand in [1] was incomplete, as was noticed by Olevskii in [5]. A few months later Guinand's claims were proved [7]. The revival of Guinand's work could be related to the discovery of quasi-crystals by Dan Shechtman (1982). The connection between crystalline measures and quasi-crystals is discussed in [6]. Up to now all examples of crystalline measures have been one-dimensional. Non-trivial two-dimensional crystalline measures are constructed in this note. It gives a simple proof of a remarkable theorem of Kurasov and Sarnak [4].
The Dirac measure located at a ∈ R n is denoted by δ a or δ a (x). A purely atomic measure is a linear combination µ = λ∈Λ c(λ)δ λ of Dirac measures, where the coefficients c(λ) are real or complex numbers and |λ|≤R |c(λ)| is finite for every R > 0. Then Λ is a countable set of points of R n . A subset Λ ⊂ R n is locally finite if Λ ∩ B is finite for every bounded set B. Equivalently, Λ can be ordered as a sequence {λ j , j = 1, 2, . . . } and |λ j | tends to infinity with j. A measure µ is a tempered distribution if it has a polynomial growth at infinity in the sense given by Laurent Schwartz in [10]. For instance, the measure The distributional Fourier transform µ of a tempered measure µ is defined by µ, φ = µ, φ , ∀φ ∈ S(R n ).
If µ is a crystalline measure, its distributional Fourier transform is also a crystalline measure. A product P µ between a crystalline measure and a finite trigonometric sum P is still a crystalline measure. If µ is a crystalline measure and A is an affine transformation of R n into itself, the image measure µ A of µ by A is still a crystalline measure. Let Γ ⊂ R n be a lattice. The distributional Fourier transform of the Dirac comb µ = vol(Γ) γ∈Γ δ γ is the Dirac comb µ = (2π) n y∈Γ * δ y on the dual lattice Γ * . Definition 1.1. Let σ j , 1 ≤ j ≤ N , be N Dirac combs, each σ j being supported by a coset x j + Γ j of a lattice Γ j ⊂ R n , 1 ≤ j ≤ N . Let P j (x) = y∈Fj c j (y) exp(2πiy · x) be a finite trigonometric sum. Let µ j = P j σ j . Then µ = µ 1 + · · · + µ N is called a generalized Dirac comb.
A generalized Dirac comb is a trivial example of a crystalline measure. A Delone set Λ ⊂ R n is defined by the following property: there exist two positive numbers R > r > 0 such that any ball of radius r, whatever its center, contains at most a point λ ∈ Λ and any ball of radius R, whatever its center, contains at least a point λ ∈ Λ. Pavel Kurasov and Peter Sarnak constructed a one-dimensional crystalline measure whose support is a Delone set [4] and which is not a generalized Dirac comb. This extends to any dimension since the tensor product µ 1 ⊗ µ 2 between two crystalline measures µ 1 and µ 2 on R is a crystalline measure on R 2 . We construct a crystalline measure σ on R 2 which is not a tensor product between two crystalline measures on R. More precisely, σ = λ∈Λ δ λ , where Λ is a Delone set.

Lemma 2.2.
There exist an integer m 0 (r) and a constant c(r) > 0 such that if m ≥ m 0 (r) and 0 ≤ k ≤ r 10 m, we have To prove (2) it suffices to estimate the right-hand side of (3) by van der Corput's lemma. The first ingredient of our construction is given by an arbitrary onedimensional crystalline measure µ. We have where Λ ⊂ R is a locally finite set of real numbers. To avoid cumbersome discussions Λ is assumed to be a Delone set and c(λ) is assumed to be a bounded sequence. Then all the series that form part of the proof of Theorem 2.5 converge in the distributional sense. The distributional Fourier transform of µ is where S ⊂ R is a locally finite set. Before constructing the two-dimensional crystalline measure σ r let us define its support M r = M r (Λ). If r ∈ (0, 1), this support is the disjoint union of two pieces. We have If r ∈ (0, 1), then M + r and M − r are two disjoint sets. Definition 2.4. With the preceding notations the atomic measure σ r is defined by This measure σ r is 1-periodic with respect to the first variable. It can also be written as Theorem 2.5. Let r ∈ [0, 1]. Then the atomic measure σ r is a crystalline measure supported by the Delone set M r .
If r = 0, this is a trivial fact, and σ 0 is a generalized Dirac comb if µ is a Dirac comb. If r = 1, Theorem 2.5 is still a trivial fact. Indeed for any λ ∈ Λ we have k∈Z (δ (k+θ1(λ),λ) + δ (k−θ1(λ),λ) ) = k∈Z (δ (k+λ,λ) + δ (k−λ,λ) ). Therefore σ 1 is a trivial crystalline measure. Indeed σ 1 is the sum of the two images of τ = k∈Z λ∈Λ c(λ)δ (k,λ) by the mappings (x 1 , x 2 ) → (x 1 ± x 2 , x 2 ). Before proving Theorem 2.5 let us consider a third example. Let α > 0 be irrational and let µ α = k∈Z δ αk . If r ∈ (0, 1), then This atomic measure is a genuine sum of Dirac measures carried by the Delone set M r . Theorem 2.5 gives a new proof of the theorem by Kurasov and Sarnak [4]. The vector space of all crystalline measures supported by M r will be described in this note (Theorem 3.3) when µ is a Dirac comb supported on αZ, where α > 1, α / ∈ Q. The proof of Theorem 2.5 is straightforward. The series used in this proof converge in the distributional sense since Λ is a Delone set and We first consider We first sum on k ∈ Z and apply the standard Poisson formula. We obtain For any m ∈ Z we consider the distribution Once this is reached we consider the corresponding series and we end with S − r = 2π m∈Z δ 2πm ⊗ g − m , where for any m ∈ Z we define a distribution We do not want to compute g ± m . Finally, We obviously have G −m = G m .
The right-hand side of (4) can be written S = ρ * µ, where µ = δ 0 ⊗ µ and is an atomic measure supported by (2πZ) 2 . This remark paves the way to a second proof of Theorem 2.5. We start with a Z 2 -periodic measure W which is given a simple geometric definition. Then we prove directly that σ r coincides with the pointwise product between the measure W and the measure dx 1 dµ(x 2 ). This pointwise product between two measures is defined in Lemma 2.10. The Fourier transform of W is computed and coincides with ρ. The Fourier transform of dx 1 dµ(x 2 ) is obviously µ. Finally, the Fourier transform of the pointwise product between W and dx 1 dµ(x 2 ) is the convolution product between ρ and µ. The support of ρ * µ is locally finite, as is checked immediately. The distributional Fourier transform of σ r is the atomic measure ρ * µ which is supported by a locally finite set. This ends the proof of Theorem 2.5.
Here are the details. Let us define W . We first consider the curve C + on the two-dimensional torus T 2 = (R/Z) 2 defined by x 1 = θ r (x 2 ). Similarly C − is defined by x 1 = −θ r (x 2 ). Let w + be the measure on C + which is the image of the measure dx 2 on T by the mapping x 2 → (θ r (x 2 ), x 2 ). Then we have Lemma 2.7. The Fourier coefficients of w + are This is obvious by the definition of the direct image of a measure. In a similar way we consider the measure w − on C − which is the image of the normalized measure dx 2 on T by the mapping x 2 → (−θ r (x 2 ), x 2 ). Then the Fourier coefficients of w − are Finally, we consider w = w + + w − . Its Fourier coefficients are (6) w(k 1 , k 2 ) = 2 1 0 cos(2πk 1 θ r (x 2 )) exp(−2πik 2 x 2 ) dx 2 .
This together with (6) implies The right-hand side of (7) is ρ(x) when ρ is defined by (5). Viewed as two Z 2 -periodic measures on R 2 the measures w ± and w are denoted by W ± and W . The support of W + is the union of the pairwise disjoint curves C + l , l ∈ Z, defined by the equations y 1 = θ r (y 2 ) + l. Similarly the support of W − is the union of the pairwise disjoint curves C − l , l ∈ Z, defined by the equations y 1 = −θ r (y 2 ) + l.
This is immediate from (5) and (7). Our second proof of Theorem 2.5 is an immediate consequence of the following lemma: Lemma 2.9. The measure σ r is the pointwise product between W and dx 1 µ(x 2 ). This pointwise product between two measures makes sense as the following lemma indicates: Lemma 2.10. Let f be a real-valued continuous function of the real variable t and let F : R → R 2 be the mapping defined by F (t) = (t, f (t)). Let l be the Lebesgue measure on R. Let ξ = F * (l) be the Radon measure on R 2 which is the pushforward of l by F . For any Radon measure µ on R the pushforward F * (µ) is the pointwise product between µ ⊗ l and ξ.
The measure µ is the weak limit of a sequence µ j of continuous functions. Then for any continuous compactly supported function φ on R 2 we have This last limit is the definition of the product between ξ and µ ⊗ l. It ends the proof of Lemma 2.10. This obvious lemma is applied to each Dirac measure δ λ , λ ∈ Γ. The product between W and dx 1 ⊗ δ λ (x 2 ) is δ (θr(λ),λ) + δ (−θr(λ),λ) . This implies Lemma 2.9. Lemma 2.9 implies that the Fourier transform of σ r is given by the convolution product between W and dx 1 ⊗ µ. We have W = ρ and dx 1 ⊗ µ = µ. Our second proof of Theorem 2.5 is complete.

Mean-periodic measures
In this section it is assumed that the atomic measure µ used in the construction of σ r is the Dirac comb on αZ, where α > 1, α / ∈ Q. Then the distributional Fourier transform ν of the crystalline measure σ r belongs to a larger class W of crystalline measures which are studied in this section. This distributional Fourier transform is a mean-periodic measure which can be calculated explicitly by simple geometric rules. This observation is proved now and paves the way to the definition of W.
Lemma 3.1 is proved in the next section. If this result is accepted, the distributional Fourier transform σ of ν makes sense. Then (10) implies τ 1 ν = 0. The zero set of τ 1 in R 2 is a collection of pairwise disjoint curves C ± l , l ∈ Z, defined by the equations y 1 = ±θ r (y 2 ) + l. Each C ± l , l ∈ Z, is contained in a vertical strip. Similarly (9) implies τ 2 ν = 0. The zero set of τ 2 is a collection of horizontal lines defined by y 2 = αq, q ∈ Z. Each of these lines is transverse to each C ± l . Therefore the distribution σ = ν is a sum of weighted Dirac measures on M r . If ν itself is a sum of weighted Dirac masses on a locally finite set, then ν and σ are necessarily two crystalline measures.
The characterization of W is given in Theorem 3.2 and completed in Theorem 3.3. We consider the vertical strip S = [−2π, 2π] × R and the rectangle R = [−2π, 2π] × [−2π/α, 2π/α]. The restriction of an atomic measure ν to a line or to a point is denoted the same way as if ν were a continuous function. Then we say that (10) holds for Theorem 3.2. Let ν 0 be an atomic measure which satisfies (9) and (11) and whose support is a locally finite set F ⊂ S. Then there exists a unique crystalline measure ν ∈ W whose restriction to S is ν 0 .
The proof is almost obvious. We rewrite (10) as an evolution equation and treat x 1 as a time variable. We obtain The initial condition is ν 0 . Then (12) is used to move from the vertical strip S = [−2π, 2π] × R to the vertical strip S 1 = [0, 4π] × R. Next we iterate to define ν on the vertical strip S m = [2(m − 1)π, 2mπ] × R, m ≥ 2. The treatment of the left vertical strips is identical. The extended atomic measure ν still satisfies (9) and (10). This ends the proof. As was stated earlier, the inverse Fourier transform σ of an atomic measure ν defined by Theorem 3.2 is a crystalline measure supported by M r .
Here is a slight improvement on Theorem 3.2. We start from an arbitrary atomic measure ν R carried be a finite subset of R = [−2π, 2π] × [−2π/α, 2π/α]. Does there exist a unique crystalline measure ν ∈ W whose restriction to R is ν R ? Here is the answer. We first construct an atomic measure ν 0 enjoying the following three properties: (a) ν 0 is carried by S = [−2π, 2π] × R, (b) ν 0 satisfies (9) on S and (11), and (c) the restriction of ν 0 to R is ν R . Once ν 0 is constructed the crystalline measure ν ∈ W is defined as above. The only obstruction to the existence of ν ∈ W is the construction of ν 0 . There are two issues. The first obstruction comes from the periodicity of ν 0 given by (9). It forces the restriction ν R (x 1 , π/α) of ν R to the upper horizontal side of R to be identical to the restriction ν R (x 1 , −π/α) of ν R to the lower horizontal side of R. Then ν R is the restriction to R of a unique atomic measure ν 0 satisfying (9). It remains to check (11). This provides the second obstruction. Indeed let us consider the restrictions of ν R to the two vertical sides of R and to {0} × [−2π/α, 2π/α]. These three atomic measures are extended by the periodicity defined by (9) and the corresponding atomic measures are denoted by θ, η, and κ. These three measures shall satisfy (11). This reads θ(x 2 ) + η(−2π, x 2 ) = rκ(x 2 + 2π) + rκ(x 2 − 2π).
We can conclude Theorem 3.3. Let ν R be an atomic measure supported by a finite subset of the rectangle R = [−2π, 2π] × [−2π/α, 2π/α]. Let us assume that ν R satisfies (11) and that ν R (x 1 , π/α) = ν R (x 1 , −π/α). Then ν R is the restriction to R of a unique crystalline measure ν ∈ W. The inverse Fourier transform of ν is a crystalline measure σ supported by M r . Conversely, any crystalline measure σ supported by M r is obtained by this construction.

Proof of Lemma 3.1
Let ν be a Radon measure satisfying (9) and (10). If g is a compactly supported continuous function, then the convolution product f = ν * g is a continuous function satisfying (9) and (10). It suffices to prove that f is a tempered distribution to conclude. This is implied by the following lemma: Lemma 4.1. There exists a constant C such that for every y ∈ R 2 and for every continuous solution f of (9) and (10) we have The Fourier series of the 2π/α-periodic continuous function x 2 → f (x 1 , x 2 ) is denoted by k a k (x 1 ) exp(iαkx 2 ). Then for any x 1 Parseval's theorem implies k |a k (x 1 )| 2 = α/2π π/α −π/α |f (x 1 , x 2 )| 2 dx 2 . Next Fubini's theorem yields We now plug (10) into the Fourier expansion f (x 1 , x 2 ) = k a k (x 1 ) exp(iαkx 2 ).
Then for any k ∈ Z we obtain 2r cos(2πkα) a k (x 1 ) = a k (x 1 + 2π) + a k (x 1 − 2π). Therefore a k (x 1 ) is a mean-periodic function of x 1 and can be expanded as a generalized Fourier series [2]. The possibly complex frequencies which appear in this expansion are the roots u ∈ C of cos(2πu) − r cos(2πkα) = 0. Since 0 < r < 1 these roots are the real numbers λ m ∈ T defined by λ m = m ± θ r (kα), m ∈ Z. This implies that a k (x 1 ) = exp(iθ r (kα)x 1 )b k (x 1 ) + exp(−iθ r (kα)x 1 )c k (x 1 ) where b k and c k are two 2π-periodic continuous functions. Since θ r − 1/4 ∞ < 1/2 the set T is uniformly discrete. Therefore there exists a constant C such that for any y 1 ∈ R we have where C does not depend on k or y 1 . Finally, (15) and (14) imply (13) when y = (y 1 , 0). This yields (13) in full generality since f is periodic in the second variable.