Stable polynomials and crystalline measures

Explicit examples of {\bf positive} crystalline measures and Fourier quasicrystals are constructed using pairs of stable of polynomials, answering several open questions in the area.


Introduction
Our investigation of the additive structure of the spectrum of metric graphs [14] provides exotic crystalline measures, in fact ones that give answers to a number of open problems. In this note we explicate the simplest examples and place the construction into the natural general setting of stable polynomials in several variables.
We recall the definitions.

Definition.
A crystalline measure µ on R is a tempered distribution of the form where δ ξ is a delta mass at ξ, and Λ and S are discrete subsets of R [25].
The basic example of a crystalline measure, in fact a Fourier quasicrystal, comes from the Poisson summation formula: (2) µ = m∈Z δ m ⇒μ = s∈Z δ 2πs , and its extension to finite combinations of these called "generalized Dirac combs" [25]. Various examples of crystallline measures that are not Dirac combs were constructed by Guinand [9]. Note however that his example 4 page 264 coming from the explicit formula in the theory of primes does not give a Fourier quasicrystal, even assuming the Riemann hypothesis. Towards a classification theory of crystalline measures µ there are a series of results that ensure that µ is a generalized Dirac comb ( [4,20,22,24]), one of the first being Theorem (Meyer [24]). If a λ take values in a finite set and |μ| is translation bounded, that is sup x∈R |μ|(x + [0, 1]) < ∞, then µ is a generalized Dirac comb.
Examples of varying complexity of Fourier quasicrystals which are not generalized Dirac combs, have been given ( [11,21,25,27]), showing that any such classification is probably very difficult [5].
A basic question which has been open for some time is whether there are positive (that is with a λ ≥ 0) crystalline measures which are not generalized Dirac combs? The constructions in Sections 2 and 3 yield such µ's which enjoy some other properties which resolve related open problems.
In Section 2 we review the definition of stable polynomials and use them to construct positive Fourier quasicrystals. In Section 3 we examine the simplest non-trivial example and use Liardet's proof of Lang's conjecture in dimension two [16,23] to analyze the additive structure of Λ, see Theorem 3. This example is rich enough for the purposes of this note. We end the section by recording the general additive structure theorem from [14] which applies to the supports Λ of the Fourier quasicrystal measures µ that are constructed from stable polynomials.

Summation formula
Stable polynomials. If P (z) = P (z 1 , . . . , z n ) is a multivariable polynomial with complex coefficients, we say that P is D = {z : |z| < 1} stable if P (z) = 0 for z = (z 1 , z 2 , . . . , z n ) with z j ∈ D for all j. To define a stable pair, consider the involution operation on P obtained by z j → 1/z j for j = 1, 2, . . . , n, the result being denoted by P ι .
Definition. Two multivariate polynomials P, Q are said to form stable pair if (1) both polynomials P and Q are D-stable; (2) there exist an integer-valued vector = ( 1 , 2 , . . . , n ) ∈ N n and a constant η such that P and Q satisfy the functional equation If such and η exists they are unique.
2) Lee-Yang pairs ([28, Theorem 5.12]) Let −1 ≤ A ij ≤ 1, A ij = A ji and where we use multi-index notation for z S = j∈S z j , the sum is over all subsets S of {1, 2, . . . , n} and S is the complement of S. Then P is a self dual stable pair For generalizations of these see [2,19,30].
For the rest of this section we show how to attach to a stable pair and real numbers b 1 , b 2 , . . . , b n > 1 a crystalline measure.
Dirichlet series. Let b 1 , b 2 , . . . , b n be real numbers larger than 1 and let ξ j = ln b j > 0, j = 1, 2, . . . , n. Let us denote by Γ + and L + the corresponding multiplicative and additive semigroups The elements of these semigroups will be denoted by b and ξ respectively b ∈ Γ + , ξ ∈ L + .
Let us introduce the following two entire functions of order 1 The stability conditions on P and Q ensure that all zeroes of F (s) and G(s) are on the imaginary axis (s) = 0. Moreover (15) implies that the zeroes for F an G are obtained from each other via reflection. F and G are finite Dirichlet series, that is Logarithmic derivatives. For (s) large enough the series for log F (s) converges absolutely: Hence for (s) 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 Note that this relation is independent of the parameter η appeared first in (3).
Logarithmic derivative as a distribution. Let Ψ ∈ C ∞ 0 (R >0 ) and Ψ(s) is entire and is rapidly decreasing when |t| → ∞ for s = σ + it, σ 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 (s) = −R picking up the residues, which areΨ(ρ), since the functionΨ is integrated with the logarithmic derivative. Summing over all zeroes of F (which are lying on the imaginary axis) we obtain Formula (20) together with expansion (19) then imply Comparing two formulas for I (expressions (23) and (25)) we may calculate the sum over the zeroes of F Summation formula. We make change of variables: We have in particular: whereĥ is the Fourier transform of h and 1 Then formula (26) becomes the following summation formula which is valid for anyĥ ∈ C ∞ 0 (R), and extends to all of S(R) as shown in the proof of Theorem 1 below. 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 S P of µ is a subset of (with L + introduced in (13)) and the Fourier transform of µ can be written as Theorem 1. 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.
Proof. The support of µ is given by the zeroes iγ j of the entire function F in (14) and hence the support Λ of µ is discrete. The support S ofμ is a subset of L + ∪ −L + ∪ {0} which is also discrete. Since m(λ) ≥ 1 and µ is positive, applying the summation formula to φ(y) is translation bounded and in particular µ and henceμ are both tempered. This shows that µ is a crystalline measure. To show that it is a Fourier quasicrystal we need to show in addition that |μ| is tempered (since µ = |µ|). To this end we first bound the coefficients c P (k) in (11). The series in (11) converges absolutely and uniformly for z in compact subsets of D n = D × D × · · · × D, and yields log P (z) = ln |P (z)| + i arg P (z) where the arg is gotten by continuous variation along the path {sz}, 0 ≤ s ≤ 1. Since P (sz), as a function of s is a polynomial in s of degree deg P , it follows that Introducing the notation e iθ = (e iθ1 , e iθ2 , . . . , e iθn ) we have from (11) that for 0 ≤ r < 1 In particular for k = 0 (33)  (30) and (34) we deduce From (29), it follows that the measure |μ| satisfies Hence |μ|([−A, A]) grows at most polynomially (∼ A n+1 ) and therefore determines a tempered distribution.
To complete the proof we invoke Theorem 11 of [8] which asserts that our translation bounded µ which has countable spectrum is an almost periodic measure in the sense of [25,Definition 5]. Remarks.
• Starting with the function G instead of F we get a similar summation formula (37) Summing the two formulas we get (38) • In the self dual case P (z) = Q(z) the summation formula takes the simplest form • The simplest stable polynomial is which is nothing else than the classical Poisson summation formula (properly scaled) (see (2) below).

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 [14]. With 1 = 1, 2 = 2 and η = −1 we get The polynomial is D-stable since the equation P (z 1 , z 2 ) = 0 can be writen as and the Möbius transformation z 1 → z1−3 1−3z1 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 T = (z 1 , z 2 ) ∈ C 2 : |z 1 | = |z 2 | = 1 . Introducing notations z 1 = e ix , z 2 = e iy 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 L(x, y) = 3 sin( and is plotted in Figure 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 ξ j = ln b j > 0. In other words, zeroes of F are situated at the intersection points between the line (γξ 1 , γξ 2 ) 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. γ 0 = 0 is always a solution since L(0, 0) = 0. All other zeroes γ j 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 where • γ j are solutions to the secular equation (43); • c(n 1 , 2n 2 ) are given by (43); • h ∈ C ∞ 0 (R) -an arbitrary test function. The difference between formula (44) and the general formula (27) is due to the fact that the stable polynomials depend just on z 2 2 . Both series on the left and right hand sides are infinite but they have different properties depending on whether ξ 1 and ξ 2 are rationally dependent or not. This is related to the number of intersection points on the torus. Also the number of zeroes iγ j is always infinite, the number of intersection points on the torus may be finite. Indeed, if ξ1 ξ2 ∈ Q, then the line is periodic on the torus, implying that there are finitely many intersection points (on the torus). The points γ j form a periodic sequence implying that 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 ξ 1 and ξ 2 are rationally independent By Kronecker's theorem the line covers the torus densely and therefore the intersection points (γ j ξ 1 , γ j ξ 2 ) cover densely the zero curve of L as well. We are interested in the rational dependence of γ j , j ∈ Z. In particular we shall need the following Lemma 1. If ξ 1 and ξ 2 are rationally independent, then the secular equation (43) L(γξ 1 , γξ 2 ) = 0 has infinitely many rationally independent solutions, i.e.
where L Q denotes the linear span with rational coefficients and dim Q the dimension of the vector space with respect to the field Q.
Proof. Assume that the dimension is finite. This means that there exists a certain M ∈ N such that every γ j for arbitrary j can be written as a rational combination of γ 1 , . . . , γ M : or equivalently Consider the multiplicative subgroup of (C * ) 2 generated by with the multiplication carried out coordinate wise. Then points (b ikj 1 , b ikj 2 ) belong to the division group Γ of Γ, that is Γ = z ∈ (C * ) 2 : z m ∈ Γ for some m ≥ 1 .
In accordance with S. Lang's conjecture [16] intersection between any algebraic subvariety and the division group for a finitely generated subgroup is along a finitely many subtori. The following theorem is proven in [23] Theorem (Liardet [23]). Assume that: • Γ is a finitely generated subgroup of the multiplicative group of the complex torus (C * ) 2 ; • Γ is the division group of Γ; • V ⊂ (C * ) 2 is an algebraic subvariety given by the zero set of Laurent polynomials. Then the intersection of V and Γ belongs to the union of a finitely many translates of certain subtori T 1 , . . . , T ν contained in V : 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 Figure 1) and the union of T j in (49) is also finite. This contradicts the fact that the number of intersection points is infinite if ξ 1 and ξ 2 are rationally independent, which completes the proof.
Our main result can be formulated as Theorem 2. For ξ 1 /ξ 2 / ∈ Q, the Fourier quasicrystal measure µ corresponding to P in (41), satisfies: i) a λ = 1 for λ ∈ Λ, that is µ is a positive "idempotent". ii) dim Q Λ = ∞, dim Q S = 2, in particular µ is not a generalized Dirac comb. iii) Λ meets any arithmetic progression in R in a finite number of points. iv) Λ is a Delone set (that is the minimal distance between elements of Λ is bounded below by a positive constant and Λ is relative dense in R) while S is not a Delone set. v) |μ| is not translation bounded.
Proof. That µ is a Fourier quasicrystal follows from Theorem 1. Note however that the argument with c(n 1 , 2n 2 ) being Fourier coefficients for log P on the torus is especially transparent, since P is real on T 2 and log P has just logarithmic singularities on the smooth curve L(x, y) = 0 and therefore is absolutely integrable. i) All zeroes of the secular equation (43) have multiplicity one and form a discrete set, hence by construction a λ = 1 and µ is a positive idempotent discrete measure. ii) Since ξ 1 /ξ 2 / ∈ Q Lemma 1 implies that dim Q Λ P = ∞, 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 S P -the support ofμ -belongs to and its dimension is equal to 2. iii) Assume that there exists a full arithmetic progression, say γ * n which intersects support of µ at an infinite number of points. Consider the corresponding group generated by (b γ * 1 , b γ * 2 ). Its intersection with the algebraic subvariety P (z 1 , z 2 ) = 0 (where P is given by (41)) is along a finitely many subtori (Liardet's Theorem) as before. The zero set contains no one-dimensional subtori, 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 ξ 1 /ξ 2 / ∈ Q. 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 page 417 in [18]). iv) 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 (ξ 1 γ, ξ 2 γ) and the zero curve of L(x, y) is non-tangential we conclude that there is a minimal distance between the consecutive solutions γ j of the secular equation (43). The function L(γξ 1 , γξ 2 ) is given by a sum of two sinus functions with amplitudes 3 and 1 implying that every interval [n 2π ] contains a solution to the secular equation. It follows that the support of µ is relatively dense and uniformly discrete, i.e. is a Delone set. The spectrum S P is not a Delone set, since otherwise the measure µ would be a generalized Dirac comb [20]. v) Similarly |μ| is not translation bounded since otherwise this would contradict Meyer's Theorem stating that every crystalline measure with a λ from a finite set (a λ = 1 in our case) and |μ| translation bounded is a generalized Dirac comb (see Introduction and [24]).

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 [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 torus 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: (A) The last question in [25]): a positive crystalline measure which is not a generalized Dirac comb; (B) Part 3 of question 11.2 in [22]: a positive Fourier quasicrystal for which every arithmetic progression meets the support in a finite set; (C) The question on page 3158 of [25] and part 2 of question 11.2 in [22]) a Fourier quasicrystal for which the support (that is Λ) is a Delone set, but the spectrum (that is S) is not; (D) Problem 4.4 in [15]: a discrete set (that is Λ) which is a Bohr almost periodic Delone set, but is not an ideal crystal.
In our forthcoming paper [14] we use higher dimensional quantitative theorems from Diophantine analysis [6,7,17,29] to show that general crystalline µ constructed in Section 2 using a stable pair P, Q with parameters b 1 , . . . , b n , satisfies: Theorem 3. For such a µ we have that i) Λ = L 1 L 2 · · · L ν N, with L 1 , . . . , L ν full arithmetic progressions and N if not empty is infinite dimensional over Q (the union means counted with multiplicities). ii) a λ take values in a finite set of positive integers; µ is a positive Fourier quasicrystal. iii) dim Q S = dim Q {ξ 1 , . . . , ξ n } . iv) There is c = c(P ) < ∞ such that any arithmetic progression in R + meets N in at most c(P ) points.

Remarks to Theorem 3:
• Theorem 3 allows us to make µ's for which dim Q S is as large as we please, however in as much as any positive crystalline measure is (measure) almost periodic it follows from Lemma 5 of [25] that S ∩ (0, ∞) or S ∩ (−∞, 0) cannot be linearly independent over Q.