Explicit examples of positive crystalline measures and Fourier quasicrystals are constructed using pairs of stable polynomials, answering several open questions in the area.
Our investigation of the additive structure of the spectrum of metric graphs1 provides exotic crystalline measures, in fact, the ones that give answers to a number of open problems. In this article, we explicate the simplest examples and place the construction into the natural general setting of stable polynomials in several variables.
We recall the definitions.
The basic example of a crystalline measure, in fact, a Fourier quasicrystal, comes from the Poisson summation formula
and its extension to finite combinations of these called “generalized Dirac combs.”2 Various examples of crystalline measures that are not Dirac combs were constructed by Guinand.4 Note, however, that his Example 4 (p. 264) coming from the explicit formula in the theory of primes does not give a Fourier quasicrystal, even assuming the Riemann hypothesis.
(Ref. 7). If aλ takes values in a finite set and is translation bounded, that is, , then μ is a generalized Dirac comb.
A basic question that has been open for some time is whether there are positive (that is, with aλ ≥ 0) crystalline measures that are not generalized Dirac combs. The constructions in Secs. II and III yield such μ’s, which enjoy some other properties, which resolve the related open problems.
In Sec. II, we review the definition of stable polynomials and use them to construct positive Fourier quasicrystals. In Sec. III, we examine the simplest non-trivial example and use Liardet’s proof of Lang’s conjecture in dimension two12,13 to analyze the additive structure of Λ (see Theorem III). This example is rich enough for the purposes of this article. We end the section by recording the general additive structure theorem from Ref. 1, which applies to the supports Λ of the Fourier quasicrystal measures μ that are constructed from stable polynomials.
II. SUMMATION FORMULA
A. Stable polynomials
If is a multivariable polynomial with complex coefficients, we say that P is stable if P(z) ≠ 0 for z = (z1, z2, …, zn) with for all j. To define a stable pair, consider the involution operation on P obtained by zj → 1/zj for j = 1, 2, …, n, the result being denoted by Pι.
Two multivariate polynomials P, Q are said to form a stable pair if
both polynomials P and Q are -stable,
- there exists an integer-valued vector and a constant η such that P and Q satisfy the functional equation(3)
- the normalization conditionis fulfilled.
If such ℓ and η exist, they are unique.
Such stable pairs arise in many contexts and there are powerful techniques for proving stability.14,15 We point to two basic examples.
- Spectral pairs. These come up as secular polynomials in quantum graphs.16,17 Let P1, P2, …, Pk be monomials in z1, …, zn of the formLet which we assume being positive for every ν = 1, …, n. If S is a k × k unitary matrix, set(4)Then, it is easy to see that(5)is a stable pair with ℓ = (ℓ1, …, ℓk) and η = (det(−S))−1.
Our studies in Ref. 1 were inspired by the trace formula for metric graphs.18–21 In fact, the presented construction via stable polynomials grew up as an attempt to understand that trace formula from a more abstract point of view.
For the rest of this section, we show how to attach to a stable pair and real numbers b1, b2, …, bn > 1, a crystalline measure.
Assume that P, Q is a stable pair of multivariable polynomials,
where MP,Q are finite subsets of
Taking the logarithm, we get the following expansion:
where, for ,
Similar formulas hold for log Q(z).
C. Dirichlet series
Let b1, b2, …, bn be real numbers larger than 1, and let ξj = ln bj > 0, j = 1, 2, …, n.
Let us denote by Γ+ and L+ the corresponding multiplicative and additive semigroups,
respectively, The elements of these semigroups will be denoted by b and ξ, respectively,
Let us introduce the following two entire functions of order 1:
The functions are related via the functional equation
where ℓ = (ℓ1, ℓ2, …, ℓn).
The stability conditions on P and Q ensure that all zeroes of F(s) and G(s) are on the imaginary axis Moreover, (15) implies that the zeroes for F and G are obtained from each other via reflection.
F and G are finite Dirichlet series, that is,
D. Logarithmic derivatives
For large enough, the series for log F(s) converges absolutely,
Hence, for large,
A similar analysis can be applied to the entire function G(s), leading to
Formula (15) establishes the following relation between the logarithmic derivatives of F and G:
for large. Note that this relation is independent of the parameter η that appeared first in (3).
E. Logarithmic derivative as a distribution
is entire and is rapidly decreasing when |t| → ∞ for s = σ + it, with σ being fixed.
Consider the integral
which is converging for large real R. We next calculate I in two different ways using the functional equation connecting F and G.
Expansion (18) gives us
To get the second representation, we shift the contour for the integral defining I to picking up the residues, which are , since the entire function is integrated with the logarithmic derivative , which is meromorphic. Summing over all zeroes of F (which are lying on the imaginary axis), we obtain
where the summation respects the multiplicity of the zeroes, and hence,
F. Summation formula
We make a change of variables,
for a certain We have, in particular,
where ĥ is the Fourier transform of h and
Then, formula (26) becomes the following summation formula:
which is valid for any and extends to all of , as shown in the Proof of Theorem 1.
To be precise, introducing the discrete support set,
obtained from the zero set of F (all lying on the imaginary axis), we define the discrete measure associated with the left-hand side of (27),
where m(λ) is the multiplicity of the corresponding zero.
Then, the spectrum SP of μ is a subset of
[with L+ introduced in (13)], and the Fourier transform of μ can be written as
Given any pair P, Q of stable polynomials satisfying assumptions (1) and (2), the measure μ is a positive crystalline measure, in fact, a Fourier quasicrystal, and is an almost periodic measure.
To complete the proof, we invoke Theorem 11 of Ref. 24, which asserts that our translation bounded μ that has a countable spectrum is an almost periodic measure in the sense of Ref. 2 (Definition 5).□
- Starting with the function G instead of F, we get a similar summation formulaSumming the two formulas, we get(37)(38)
- In the self-dual case P(z) = Q(z), the summation formula takes the simplest form(39)
III. THE FIRST NON-TRIVIAL EXAMPLE
Our goal in this section is to present an explicit example of a positive crystalline measure. Consider the following polynomial:
in fact, describing the non-linear part of the spectrum of the lasso graph.1 With ℓ1 = 1, ℓ2 = 2, and η = −1, we get
The polynomial is -stable since the equation P(z1, z2) = 0 can be writen as
and the Möbius transformation maps the unit disk to its complement.
The Dirichlet series is equal to
To determine the zero set of F(s), let us first describe the zero set of P on the unit torus Introducing notations z1 = eix, z2 = eiy, the same torus can be seen as the square [0, 2π] × [0, 2π] with the opposite sides identified.
Then, the zero set is described by the Laurent polynomial
and is plotted in Fig. 1.
Note that the normal to the curve always lies in the first quadrant, in fact,
where we used that L(x, y) = 0.
Knowing the zero set of L(x, y), the zeroes of the Dirichlet series F(s) (all lying on the imaginary axis) are obtained in the following way:
where we used that ξj = lnbj > 0. In other words, zeroes of F are situated at the intersection points between the line (γξ1, γξ2) and the zero curve for L. Both the normal to the zero curve and the guide vector for the line belong to the first quadrant; hence, the intersection is never tangential. This implies, in particular, that all zeroes are simple. γ0 = 0 is always a solution since L(0, 0) = 0. All other zeroes γj indicate the distance between the intersection points and the origin measured along the line. It is clear that L(−x, −y) = −L(x, y) [which also follows from (15) and the fact that F = G in the current example], implying that the zeroes are symmetric with respect to the origin.
The summation formula (27) takes the form
Both series on the left- and right-hand sides are infinite, but they have different properties depending on whether ξ1 and ξ2 are rationally dependent or not. This is related to the number of intersection points on the torus. The number of zeroes iγj is also always infinite, and the number of intersection points on the torus may be finite. Indeed, if , then the line is periodic on the torus, implying that there are finitely many intersection points (on the torus). The points γj form a periodic sequence, implying that the obtained summation formula is just a finite sum of Poisson summation formulas with the same period and μ is a generalized Dirac comb.
Next, we assume that ξ1 and ξ2 are rationally independent,
By Kronecker’s theorem, the line covers the torus densely, and therefore, the intersection points (γjξ1, γjξ2) cover densely the zero curve of L as well. We are interested in the rational dependence of In particular, we shall need the following:
(Ref. 13). Assume that
Γ is a finitely generated subgroup of the multiplicative group of the complex torus
is the division group of Γ, and
is an algebraic subvariety given by the zero set of Laurent polynomials.
Now, no line belongs to the zero set of L, so L contains no one-dimensional subtori, and hence, the intersection of the zero set (the curves plotted in Fig. 1) and the union of Tj in (49) is also finite being the intersection of V with finitely many zero-dimensional subtori. This contradicts the fact that the number of intersection points is infinite if ξ1 and ξ2 are rationally independent, which completes the proof.□
Our main result can be formulated as follows:
For , the Fourier quasicrystal measure μ corresponding to P in (41) satisfies as follows:
aλ = 1 for λ ∈ Λ, that is, μ is a positive “idempotent.”
in particular, μ is not a generalized Dirac comb.
Λ meets any arithmetic progression in in a finite number of points.
Λ is a Delone set (that is, the minimal distance between elements of Λ is bounded below by a positive constant and Λ is relatively dense in ), while S is not a Delone set.
is not translation bounded.
That μ is a Fourier quasicrystal follows from Theorem 1. Note, however, that the argument with c(n1, 2n2) being Fourier coefficients for log P on the torus is especially transparent, since P is real on and log P has just logarithmic singularities on the smooth curve L(x, y) = 0 and, therefore, is absolutely integrable.
All zeroes of the secular Eq. (43) have multiplicity one and form a discrete set; hence, by construction, aλ = 1 and μ is a positive idempotent discrete measure.
- Since , Lemma 1 implies that ; hence, the support of μ is not contained in a finite union of translates of any lattice and μ is not a generalized Dirac comb. The spectrum SP—the support of —belongs toand its dimension is equal to 2.
Assume that there exists a full arithmetic progression, say, γ*n, which intersects the support of μ at an infinite number of points. Consider the corresponding group generated by Its intersection with the algebraic subvariety P(z1, z2) = 0 [where P is given by (41)] is along finitely many subtori (Liardet’s theorem) as before. The zero set contains no one-dimensional subtori, and hence, the number of intersection points on the torus is finite. The number of intersection points between the arithmetic progression and the zero set can be infinite only if certain points occur several times, but this is impossible since It follows that the intersection between any arithmetic progression and Λ is always finite. The same result could be proven using Lech’s theorem (lemma on p. 417 in Ref. 25).
The zero set of L(x, y) is given by two non-intersecting curves on [0,2π]2, implying that there is a minimal distance ρ between the different components of the curve. Taking into account that the intersection between the line (ξ1γ, ξ2γ) and the zero curve of L(x, y) is non-tangential, we conclude that there is a minimal distance between the consecutive solutions γj of the secular Eq. (43). The function L(γξ1, γξ2) is given by a sum of two sinus functions with amplitudes 3 and 1 implying that every interval contains a solution to the secular equation. It follows that the support of μ is relatively dense and uniformly discrete, i.e., it is a Delone set. The spectrum SP is not a Delone set, since the measure μ would be a generalized Dirac comb.6
Similarly, is not translation bounded, since this would contradict Meyer’s theorem stating that every crystalline measure with aλ from a finite set (aλ = 1 in our case) and translation bounded is a generalized Dirac comb (see Sec I and Ref. 7).□
Remarks to Theorem 2:
Properties (ii) and (iii) show that the measures μ in the theorem are far from being generalized Dirac combs.
In Theorem 5.16 of Ref. 26, a positive measure μ of the type in (1) is constructed for which Λ is discrete but for which S need not be (called there a Poisson measure). In fact, these Λ’s can be realized as the intersection of the graph of a periodic continuous function on the two tori with an irrational line and as such are of a similar shape to our μ’s.
The measures μ in Theorem 2 provide examples answering the following questions concerning crystalline measures:
The last question in Ref. 2:
a positive crystalline measure which is not a generalized Dirac comb;
Part 3 of question 11.2 in Ref. 3:
a positive Fourier quasicrystal for which every arithmetic progression meets the support in a finite set;
a Fourier quasicrystal for which the support (that is Λ) is a Delone set, but the spectrum (that is S) is not;
Problem 4.4 in Ref. 27:
a discrete set (that is Λ) which is a Bohr almost periodic Delone set, but is not an ideal crystal.
In our forthcoming paper,1 we use higher dimensional quantitative theorems from Diophantine analysis28–31 to show that general crystalline μ constructed in Sec. II using a stable pair P, Q with parameters b1, …, bn, satisfies the following:
For such a μ, we have that
Λ = L1 ⊔ L2 ⊔ ⋯ ⊔ Lν ⊔ N, with L1, …, Lν full arithmetic progressions and N if not empty is infinite dimensional over (the union ⊔ means counted with multiplicities).
aλ take values in a finite set of positive integers; μ is a positive Fourier quasicrystal.
There is c = c(P) < ∞ such that any arithmetic progression in meets N in at most c(P) points.
Remarks to Theorem 3:
Theorem 3 allows us to make μ’s for which is as large as we wish however, in as much as any positive crystalline measure is (measure) almost periodic, it follows from Lemma 5 of Ref. 2 that S ∩ (0, ∞) or S ∩ (−∞, 0) cannot be linearly independent over
The authors would like to thank Boris Shapiro for initiating our collaboration, Yves Meyer for attracting our attention to crystalline measures and pointing us to his paper,26 Alexei Poltoratskii for pointing out the importance of positive crystalline measures, and Nir Lev and Alexander Olevskii for their comments.
This paper is written in memory of our brilliant colleague Jean Bourgain.