Stable polynomials and crystalline measures

Kurasov, P.; Sarnak, P.

doi:10.1063/5.0012286

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

I. INTRODUCTION

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.

Definition.

A crystalline measure μ on

R

is a tempered distribution of the form

μ = \sum_{λ \in Λ} a_{λ} δ_{λ} for which \hat{μ} = \sum_{s \in S} b_{s} δ_{s},

(1)

where δ_ξ is a delta mass at ξ and Λ and S are discrete subsets of

R

.²

If both |μ| and $| \hat{μ} |$ are tempered as well, then following Ref. 3 (Sec. 1.1), we call μ a Fourier quasicrystal.

The basic example of a crystalline measure, in fact, a Fourier quasicrystal, comes from the Poisson summation formula

μ = \sum_{m \in Z} δ_{m} \Rightarrow \hat{μ} = \sum_{s \in Z} δ_{2 π s},

(2)

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.

Toward a classification theory of crystalline measures μ, there is a series of results that ensures that μ is a generalized Dirac comb (Refs. 3 and 5–7), one of the first being the following theorem:

Theorem

(Ref. 7). If a_λ takes values in a finite set and $| \hat{μ} |$ is translation bounded, that is, $\sup_{x \in R} | \hat{μ} | (x + [0, 1]) < \infty$ , then μ is a generalized Dirac comb.

Examples of varying complexity of Fourier quasicrystals, which are not generalized Dirac combs, are given in Refs. 2 and 8–10, showing that any such classification is probably very difficult.¹¹

A basic question that has been open for some time is whether there are positive (that is, with a_λ ≥ 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 two^12,13 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 $P (z) = P (z_{1}, \dots, z_{n})$ is a multivariable polynomial with complex coefficients, we say that P is $D = \{z : | z | < 1\}$ stable if P(z) ≠ 0 for z = (z₁, z₂, …, z_n) with $z_{j} \in D$ 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 a stable pair if

both polynomials P and Q are $D$ -stable,
there exists an integer-valued vector $ℓ = (ℓ_{1}, ℓ_{2}, \dots, ℓ_{n}) \in N^{n}$ and a constant η such that P and Q satisfy the functional equation
$Q (z) = η z_{1}^{ℓ_{1}} z_{2}^{ℓ_{2}} \dots z_{n}^{ℓ_{n}} P^{ι} (z), a n d$
(3)
the normalization condition
$P (\vec{0}) = Q (\vec{0}) = 1$
is 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, …, P_k be monomials in z₁, …, z_n of the form
$P_{j} (z) = z_{1}^{a_{j, 1}} z_{2}^{a_{j, 2}} \dots z_{n}^{a_{j, n}}, j = 1, 2, \dots, k,$
(4)
$a_{j, ν} \in N .$ Let $ℓ_{ν} = \sum_{j = 1}^{k} a_{j, ν},$ which we assume being positive for every ν = 1, …, n. If S is a k × k unitary matrix, set
$R_{S} (z) = \det [I_{k \times k} - [\begin{bmatrix} P_{1} (z) & 0 & \dots & 0 \\ 0 & P_{2} (z) & \dots & 0 \\ ⋮ & ⋮ & ⋱ & ⋮ \\ 0 & 0 & \dots & P_{k} (z) \end{bmatrix}] S] .$
(5)
Then, it is easy to see that
$P = R_{S}, Q = R_{S^{- 1}}$
is a stable pair with ℓ = (ℓ₁, …, ℓ_k) 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.
Lee-Yang pairs (Ref. 22, Theorem 5.12). Let −1 ≤ A_ij ≤ 1, A_ij = A_ji, and
$P (z) = \sum_{S} (\prod_{i \in S} \prod_{j \in S^{'}} A_{i j}) z^{S},$
(6)
where we use a multi-index notation for z^S = ∏_j∈Sz_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,
$(z_{1} z_{2} \dots z_{n}) P^{ι} (z) = P (z) .$
(7)
For generalizations of these, see Refs. 14, 15, and 23.

For the rest of this section, we show how to attach to a stable pair and real numbers b₁, b₂, …, b_n > 1, a crystalline measure.

B. Notations

Assume that P, Q is a stable pair of multivariable polynomials,

P (z) = 1 + \sum_{m \in M_{P}} a_{P} (m) z^{m}, Q (z) = 1 + \sum_{m \in M_{Q}} a_{Q} (m) z^{m},

(8)

where M_P,Q are finite subsets of

Z_{+}^{n} ≔ \{k = (k_{1}, k_{2}, \dots, k_{n}), k_{j} \in Z, k_{j} \geq 0, k \neq (0, 0, \dots, 0)\} .

(9)

Taking the logarithm, we get the following expansion:

\begin{aligned} \log P (z) & = \sum_{ν = 1}^{\infty} \frac{{(- 1)}^{ν}}{ν} {(\sum_{m \in M_{P}} a_{P} (m) z^{m})}^{ν} \\ = \sum_{ν = 1}^{\infty} \frac{{(- 1)}^{ν}}{ν} \sum_{m_{1}, m_{2}, \dots, m_{ν} \in M_{P}} a_{P} (m_{1}) a_{P} (m_{2}) \dots a_{P} (m_{ν}) z^{m_{1}} z^{m_{2}} \dots z^{m_{ν}} \\ = \sum_{ν = 1}^{\infty} \frac{{(- 1)}^{ν}}{ν} \sum_{k \in Z_{+}^{n}} \sum_{m_{1} + m_{2} + \dots + m_{ν} = k} a_{P} (m_{1}) a_{P} (m_{2}) \dots a_{P} (m_{ν}) z^{k}, \end{aligned}

(10)

and hence,

\log P (z) = \sum_{k \in Z_{+}^{n}} c_{P} (k) z^{k},

(11)

where, for $k \in Z_{+}^{n}$ ⁠,

c_{P} (k) = \sum_{ν = 1}^{\sum_{j = 1}^{n} k_{j}} \sum_{m_{1} + m_{2} + \dots + m_{ν} = k} \frac{{(- 1)}^{ν} a_{P} (m_{1}) a_{P} (m_{2}) \dots a_{P} (m_{ν})}{ν} .

(12)

Similar formulas hold for log Q(z).

C. Dirichlet series

Let b₁, b₂, …, 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,

\begin{align} Γ_{+} & = \{b_{1}^{m_{1}} b_{2}^{m_{2}} \dots b_{n}^{m_{n}} : m_{j} \in N \cup {0}\} \ {1}, \\ L_{+} & = \log Γ_{+} \\ = \{m_{1} ξ_{1} + m_{2} ξ_{2} + \dots + m_{n} ξ_{n} : m_{j} \in N \cup {0}\} \ {0}, \end{align}

(13)

respectively, The elements of these semigroups will be denoted by b and ξ, respectively,

b \in Γ_{+}, ξ \in L_{+} .

Let us introduce the following two entire functions of order 1:

\begin{aligned} F (s) & ≔ P (b_{1}^{- s}, b_{2}^{- s}, \dots, b_{n}^{- s}) \equiv P (e^{- ξ_{1} s}, e^{- ξ_{2} s} \dots, e^{- ξ_{n} s}), \\ G (s) & ≔ Q (b_{1}^{- s}, b_{2}^{- s}, \dots, b_{n}^{- s}) \equiv Q (e^{- ξ_{1} s}, e^{- ξ_{2} s} \dots, e^{- ξ_{n} s}), \\ s \in C . \end{aligned}

(14)

The functions are related via the functional equation

\begin{aligned} F (- s) & = P (b_{1}^{s}, b_{2}^{s}, \dots, b_{n}^{s}) \\ = P^{ι} (b_{1}^{- s}, \dots, b_{n}^{- s}) \\ = η^{- 1} {(b_{1}^{ℓ_{1}} \dots b_{n}^{ℓ_{n}})}^{s} G (s) \end{aligned}

\Rightarrow F (- s) = η^{- 1} {(b_{1}^{ℓ})}^{s} G (s),

(15)

where ℓ = (ℓ₁, ℓ₂, …, ℓ_n).

The stability conditions on P and Q ensure that all zeroes of F(s) and G(s) are on the imaginary axis $R (s) = 0 .$ 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,

\begin{aligned} F (s) = 1 + \sum_{m \in M_{P}} a_{P} (m) {(b^{m})}^{- s}, \\ G (s) = 1 + \sum_{m \in M_{Q}} a_{Q} (m) {(b^{m})}^{- s} . \end{aligned}

(16)

D. Logarithmic derivatives

For $R (s)$ large enough, the series for log F(s) converges absolutely,

\begin{aligned} \log F (s) & = \sum_{k \in Z_{+}^{n}} c_{P} (k) e^{- (k_{1} ξ_{1} + k_{2} ξ_{2} + \dots + k_{n} ξ_{n}) s} \\ = \sum_{k \in Z_{+}^{n}} c_{P} (k) e^{- (ξ \cdot k) s} . \end{aligned}

(17)

Hence, for $R (s)$ large,

\frac{F^{'} (s)}{F (s)} = - \sum_{k \in Z_{+}^{n}} (ξ \cdot k) c_{P} (k) e^{- (ξ \cdot k) s} .

(18)

A similar analysis can be applied to the entire function G(s), leading to

\frac{G^{'} (s)}{G (s)} = - \sum_{k \in Z_{+}^{n}} (ξ \cdot k) c_{Q} (k) e^{- (ξ \cdot k) s} .

(19)

Formula (15) establishes the following relation between the logarithmic derivatives of F and G:

\log F (- s) = - \log η + s (ξ \cdot ℓ) + \log G (s)

\Rightarrow - \frac{F^{'} (- s)}{F (- s)} = (ξ \cdot ℓ) + \frac{G^{'} (s)}{G (s)}

(20)

for $R (s)$ large. Note that this relation is independent of the parameter η that appeared first in (3).

E. Logarithmic derivative as a distribution

Let $Ψ \in C_{0}^{\infty} (R_{> 0})$ and

\tilde{Ψ} (s) = \int_{0}^{\infty} Ψ (x) x^{s} \frac{d x}{x} .

(21)

$\tilde{Ψ} (s)$ is entire and is rapidly decreasing when |t| → ∞ for s = σ + it, with σ being fixed.

Consider the integral

I ≔ \frac{1}{2 π i} \int_{R (s) = R} \frac{F^{'} (s)}{F (s)} \tilde{Ψ} (s) d s,

(22)

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

\begin{align} I & = \frac{1}{2 π i} \int_{R (s) = R} (- \sum_{k \in Z_{+}^{n}} (ξ \cdot k) c_{P} (k) e^{- (ξ \cdot k) s}) \tilde{Ψ} (s) d s \\ = - \sum_{k \in Z_{+}^{n}} (ξ \cdot k) c_{P} (k) \frac{1}{2 π i} \int_{R (s) = R} \tilde{Ψ} (s) e^{- (ξ \cdot k) s} d s . \end{align}

(23)

To get the second representation, we shift the contour for the integral defining I to $R (s) = - R$ picking up the residues, which are $\tilde{Ψ} (ρ)$ ⁠, since the entire function $\tilde{Ψ}$ is integrated with the logarithmic derivative $\frac{F^{'}}{F}$ ⁠, which is meromorphic. Summing over all zeroes of F (which are lying on the imaginary axis), we obtain

\sum_{ρ : F (ρ) = 0} \tilde{Ψ} (ρ),

(24)

where the summation respects the multiplicity of the zeroes, and hence,

\begin{aligned} I & = \sum_{ρ : F (ρ) = 0} \tilde{Ψ} (ρ) + \frac{1}{2 π i} \int_{R (s) = - R} \frac{F^{'} (s)}{F (s)} \tilde{Ψ} (s) d s \\ = \sum_{ρ : F (ρ) = 0} \tilde{Ψ} (ρ) + \frac{1}{2 π i} \int_{R (s) = R} \frac{F^{'} (- s)}{F (- s)} \tilde{Ψ} (- s) d s . \end{aligned}

Formula (20) together with expansion (19) then implies

\begin{aligned} I & = \sum_{ρ : F (ρ) = 0} \tilde{Ψ} (ρ) - (ξ \cdot ℓ) \frac{1}{2 π i} \int_{R (s) = R} \tilde{Ψ} (- s) d s \\ + \sum_{k \in Z_{+}^{n}} (ξ \cdot k) c_{Q} (k) \frac{1}{2 π i} \int_{R (s) = R} \tilde{Ψ} (- s) e^{- (ξ \cdot k) s} d s . \end{aligned}

(25)

Comparing two formulas for I [expressions (23) and (25)], we may calculate the sum over the zeroes of F,

\begin{align} \sum_{ρ : F (ρ) = 0} & \tilde{Ψ} (ρ) = (ξ \cdot ℓ) \frac{1}{2 π i} \int_{R (s) = R} \tilde{Ψ} (s) d s \\ - \sum_{k \in Z_{+}^{n}} (ξ \cdot k) c_{P} (k) \frac{1}{2 π i} \int_{R (s) = R} \tilde{Ψ} (s) e^{- (ξ \cdot k) s} d s \\ - \sum_{k \in Z_{+}^{n}} (ξ \cdot k) c_{Q} (k) \frac{1}{2 π i} \int_{R (s) = R} \tilde{Ψ} (- s) e^{- (ξ \cdot k) s} d s . \end{align}

(26)

F. Summation formula

We make a change of variables,

\begin{aligned} x = e^{t}, \\ (0, + \infty) \to (- \infty, + \infty) \end{aligned}

so that

Ψ (e^{t}) = h (t)

for a certain $h \in C_{0}^{\infty} (R) .$ We have, in particular,

\begin{aligned} \tilde{Ψ} (i γ) & = \int_{0}^{\infty} Ψ (x) x^{i γ} \frac{d x}{x} = [\begin{bmatrix} x = e^{t} \\ d x = e^{t} d t \end{bmatrix}] \\ = \int_{- \infty}^{\infty} h (t) e^{i γ t} d t = ĥ (γ), \end{aligned}

where ĥ is the Fourier transform of h and

\begin{aligned} \frac{1}{2 π i} \int_{R (s) = R} \tilde{Ψ} (s) e^{- (ξ \cdot k) s} d s & = \frac{1}{2 π i} \int_{R (s) = R} (\int_{0}^{\infty} Ψ (x) x^{s} \frac{d x}{x}) e^{- (ξ \cdot k) s} d s \\ = \frac{1}{2 π} \int_{- \infty}^{+ \infty} (\int_{- \infty}^{\infty} h (t) e^{(R + i s) t} d t) e^{- (ξ \cdot k) R} e^{- i (ξ \cdot k) s} d s \\ = h (ξ \cdot k) . \end{aligned}

Then, formula (26) becomes the following summation formula:

\begin{align} \sum_{γ : F (i γ) = 0} ĥ (γ) & = (ξ \cdot ℓ) h (0) - \sum_{k \in Z_{+}^{n}} (ξ \cdot k) c_{P} (k) h (ξ \cdot k) \\ - \sum_{k \in Z_{+}^{n}} (ξ \cdot k) c_{Q} (k) h (- ξ \cdot k), \end{align}

(27)

which is valid for any $ĥ \in C_{0}^{\infty} (R)$ and extends to all of $S (R)$ ⁠, as shown in the Proof of Theorem 1.

To be precise, introducing the discrete support set,

Λ_{P} ≔ {γ : F (i γ) = 0},

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),

μ = μ_{P} ≔ \sum_{γ : F (i γ) = 0} δ_{γ} \equiv \sum_{λ \in Λ_{P}} m (λ) δ_{λ},

(28)

where m(λ) is the multiplicity of the corresponding zero.

Then, the spectrum S_P of μ is a subset of

L_{+} \cup - L_{+} \cup {0}

[with L₊ introduced in (13)], and the Fourier transform of μ can be written as

\hat{μ} = (ξ \cdot ℓ) - \sum_{k \in Z_{+}^{n}} (ξ \cdot k) c_{P} (k) δ_{ξ \cdot k} - \sum_{k \in Z_{+}^{n}} (ξ \cdot k) c_{Q} (k) δ_{- ξ \cdot k} .

(29)

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 (), and hence, the support Λ of μ is discrete. The support S of

\hat{μ}

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) = ϕ₀(x − y) with ϕ₀ ≥ 0, ϕ₀ ≥ 1 on [−1, 1] and

{\hat{ϕ}}_{0}

having compact support in (−ɛ₀, ɛ₀) where (−ɛ₀, ɛ₀) ∩ (L₊ ∪ −L₊) is empty yields

\sum_{λ : x - 1 \leq λ \leq x + 1} m (λ) ≪ (ξ \cdot ℓ) {\hat{ϕ}}_{0} (0)

uniformly in x. That is, μ = |μ| is translation bounded and, in particular, μ and hence

\hat{μ}

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

| \hat{μ} |

is tempered (since μ = |μ|). To this end, we first bound the coefficients c_P(k) in (). The series in () converges absolutely and uniformly for z in compact subsets of

D^{n} = D \times D \times \dots \times D

and yields log P(z) = ln |P(z)| + i arg P(z), where the arg is obtained by continuous variation along the path {sz}, 0 ≤ s ≤ 1. P(sz) as a function of s is a polynomial in s of degree deg P with the constant term equal to P(0) = 1, and each term in the polynomial may contribute with at most π to the argument, and hence,

| \arg P (z) | \leq π (\deg P) .

(30)

Let

K = \sup_{z \in D^{n}} | P (z) |

⁠, then

\ln | P (z) | \leq \ln K for z \in D^{n} .

(31)

Introducing the notation

e^{i θ} = (e^{i θ_{1}}, e^{i θ_{2}}, \dots, e^{i θ_{n}}),

we have from () that for 0 ≤ r < 1,

\int_{T^{n}} \log P (r e^{i θ}) e^{- i k \cdot θ} d θ = r^{[) k |} c_{P} (k) .

(32)

In particular, for k = 0,

\int_{T^{n}} \ln | P (r e^{i θ}) | d θ = 0,

(33)

since the constant term is absent in (). According to (),

\ln | P (r e^{i θ}) | - \ln K \leq 0,

and hence,

\begin{align} \int_{T^{n}} |\ln | P (r e^{i θ}) || d θ & = \int_{T^{n}} |\ln | P (r e^{i θ}) | - \ln K + \ln K| d θ \\ \leq - \int_{T^{n}} (\ln | P (r e^{i θ}) | - \ln K) d θ + \ln K \\ = 2 \ln K \end{align}

(34)

by (). From () and the r independent bounds () and (), we deduce

|c_{P} (k)| \leq C < \infty .

(35)

From (), it follows that the measure

| \hat{μ} |

satisfies

\begin{align} | \hat{μ} | ([- A, A]) \leq ξ \cdot ℓ + 2 \sum_{k \in Z_{+}^{n}, ξ \cdot k \leq A} (ξ \cdot k) | c_{P} (k) | \\ \leq ξ \cdot ℓ + 2 \max {ξ_{j}} C \sum_{\overset{k \in Z_{+}^{n},}{k_{1} + k_{2} + \dots + k_{n} \leq A / \min {ξ_{j}}}} (k_{1} + k_{2} + \dots + k_{n}) \\ \leq ξ \cdot ℓ + 2 \max {ξ_{j}} C {([\frac{A}{\min {ξ_{j}}}] + 1)}^{n + 1} . \end{align}

(36)

Hence,

| \hat{μ} | ([- A, A])

grows at most polynomially (∼Aⁿ⁺¹) and, therefore, determines a tempered distribution.

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).□

G. Remarks

Starting with the function G instead of F, we get a similar summation formula
$\begin{align} \sum_{γ : F (i γ) = 0} ĥ (- γ) & = (ξ \cdot ℓ) h (0) - \sum_{k \in Z_{+}^{n}} (ξ \cdot k) c_{Q} (k) h (ξ \cdot k) \\ - \sum_{k \in Z_{+}^{n}} (ξ \cdot k) c_{P} (k) h (- ξ \cdot k) . \end{align}$
(37)
Summing the two formulas, we get
$\begin{align} \sum_{γ : F (i γ) = 0} (ĥ (γ) + ĥ (- γ)) = 2 (ξ \cdot ℓ) h (0) \\ - \sum_{k \in Z_{+}^{n}} (ξ \cdot k) (c_{P} (k) + c_{Q} (k)) (h (ξ \cdot k) + h (- ξ \cdot k)) . \end{align}$
(38)
In the self-dual case P(z) = Q(z), the summation formula takes the simplest form
$\begin{align} \sum_{γ : F (i γ) = 0} ĥ (γ) = (ξ \cdot ℓ) h (0) \\ - \sum_{k \in Z_{+}^{n}} (ξ \cdot k) c_{P} (k) (h (ξ \cdot k) + h (- ξ \cdot k)) . \end{align}$
(39)
The simplest stable polynomial is
$P (z_{1}) = 1 - z_{1} .$
For it,
$\begin{aligned} Q (z_{1}) & = z_{1} - 1, \\ F (s) & = 1 - 1 / b_{1}^{s}, \\ γ_{n} & = \frac{2 π}{ξ_{1}} n, n \in Z, \\ \log F (s) & = \log (1 - \frac{1}{b_{1}^{s}}) = - \sum_{n = 1}^{\infty} \frac{1}{n} \frac{1}{{(b_{1}^{n})}^{s}}, \\ \frac{F^{'}}{F} (s) & = \sum_{n = 1}^{\infty} \frac{1}{{(b_{1}^{n})}^{s}} ξ_{1} . \end{aligned}$
Substitution into the summation formula (27) gives
$\begin{aligned} \sum_{n \in Z} ĥ (\frac{2 π}{ξ_{1}} n) & = ξ_{1} (h (0) + \sum_{n = 1}^{\infty} (h (n ξ_{1}) + h (- n ξ_{1}))) \\ \equiv ξ_{1} \sum_{n \in Z} h (n ξ_{1}), \end{aligned}$
(40)
which is nothing else than the classical Poisson summation formula (properly scaled) [see (2)].

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:

P (z_{1}, z_{2}) = 1 - \frac{1}{3} z_{1} + \frac{1}{3} z_{2}^{2} - z_{1} z_{2}^{2},

(41)

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

Q (z_{1}, z_{2}) = (- 1) (z_{1} z_{2}^{2} - \frac{1}{3} z_{2}^{2} + \frac{1}{3} z_{1} - 1) \equiv P (z_{1}, z_{2}) .

The polynomial is $D$ -stable since the equation P(z₁, z₂) = 0 can be writen as

\frac{z_{1} - 3}{1 - 3 z_{1}} = z_{2}^{2},

and the Möbius transformation $z_{1} \mapsto \frac{z_{1} - 3}{1 - 3 z_{1}}$ maps the unit disk to its complement.

The Dirichlet series is equal to

F (s) = 1 - \frac{1}{3} \frac{1}{b_{1}^{s}} + \frac{1}{3} \frac{1}{b_{2}^{2 s}} - \frac{1}{b_{1}^{s} b_{2}^{2 s}},

\begin{aligned} \log F (s) & = - \sum_{k = 1}^{\infty} \frac{1}{k} {(\frac{1}{3} \frac{1}{b_{1}^{s}} - \frac{1}{3} \frac{1}{b_{2}^{2 s}} + \frac{1}{b_{1}^{s} b_{2}^{2 s}})}^{k} \\ = \sum_{(n_{1}, n_{2}) \in Z_{+}^{2}} c (n_{1}, 2 n_{2}) \frac{1}{b_{1}^{n_{1} s} b_{2}^{2 n_{2} s}}, \end{aligned}

with

c (n_{1}, 2 n_{2}) = - \sum_{\overset{k_{1}, k_{2}, k_{3} \in N \cup {0}}{\overset{k_{1} + k_{3} = n_{1}}{k_{2} + k_{3} = n_{2}}}} \frac{(k_{1} + k_{2} + k_{3} - 1)!}{k_{1}! k_{2}! k_{3}!} \frac{{(- 1)}^{k_{2}}}{3^{k_{1} + k_{2}}} .

(42)

To determine the zero set of F(s), let us first describe the zero set of P on the unit torus $T = \{(z_{1}, z_{2}) \in C^{2} : | z_{1} | = | z_{2} | = 1\} .$ Introducing notations z₁ = e^ix, z₂ = 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 (\frac{x}{2} + y) + \sin (\frac{x}{2} - y)

and is plotted in Fig. 1.

FIG. 1.

View large Download slide

Zero set for L(x, y).

Note that the normal to the curve always lies in the first quadrant, in fact,

\begin{aligned} \frac{\partial y}{\partial x} & = - \frac{\frac{\partial L (x, y)}{\partial x}}{\frac{\partial L (x, y)}{\partial y}} \\ = - \frac{1}{2} \frac{3 \cos (x / 2 + y) + \cos (x / 2 - y)}{3 \cos (x / 2 + 1) - \cos (x / 2 - y)} \\ = - \frac{1}{2} \frac{8}{{(3 \cos (x / 2 + y) - \cos (x / 2 - y))}^{2}} \\ < 0, \end{aligned}

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:

0 = F (i γ_{j}) = P (b_{1}^{i γ_{j}}, b_{2}^{i γ_{j}}) = P (e^{i γ_{j} ξ_{1}}, e^{i γ_{j} ξ_{2}}) = L (γ_{j} ξ_{1}, γ_{j} ξ_{2})

\Leftrightarrow 3 \sin ((\frac{ξ_{1}}{2} + ξ_{2}) γ_{j}) + \sin ((\frac{ξ_{1}}{2} - ξ_{2}) γ_{j}) = 0,

(43)

where we used that ξ_j = lnb_j > 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 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

\begin{aligned} \sum_{γ_{j}} ĥ (γ_{j}) = (ξ_{1} + 2 ξ_{2}) h (0) & - \sum_{n = (n_{1}, n_{2}) \in Z_{+}^{2}} c (n_{1}, 2 n_{2}) (n_{1} ξ_{1} + 2 n_{2} b_{2}) (h (n_{1} ξ_{1} + 2 n_{2} ξ_{2})) \\ (+ h (- (n_{1} ξ_{1} + 2 n_{2} ξ_{2}))), \end{aligned}

(44)

where

γ_j are solutions to the secular Eq. (43),
c(n₁, 2n₂) are given by (42), and
$h \in C_{0}^{\infty} (R)$ is an arbitrary test function.

The difference between formula (44) and the general formula (27) is due to the fact that the stable polynomials just depend on $z_{2}^{2}$ ⁠.

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 iγ_j is also always infinite, and the number of intersection points on the torus may be finite. Indeed, if $\frac{ξ_{1}}{ξ_{2}} \in Q$ ⁠, 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 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,

\frac{ξ_{1}}{ξ_{2}} \notin Q .

(45)

By Kronecker’s theorem, the line covers the torus densely, and therefore, the intersection points (γ_jξ₁, γ_jξ₂) cover densely the zero curve of L as well. We are interested in the rational dependence of $γ_{j}, j \in Z .$ In particular, we shall need the following:

Lemma 1.

If ξ₁ and ξ₂ are rationally independent, then the secular Eq. (),

L (γ ξ_{1}, γ ξ_{2}) = 0,

has infinitely many rationally independent solutions, i.e.,

\dim_{Q} L_{Q} {γ_{j}}_{j \in Z} = \infty,

(46)

where

L_{Q}

denotes the linear span with rational coefficients and

\dim_{Q}

denotes 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 γ₁, …, γ_M,

γ_{j} = a_{1}^{j} γ_{1} + a_{2}^{j} γ_{2} + \dots + a_{M}^{j} γ_{M}, a_{m}^{j} \in Q .

(47)

It follows that

e^{i γ_{j} ξ_{α}} = {(e^{i γ_{1} ξ_{α}})}^{a_{1}^{j}} {(e^{i γ_{2} ξ_{α}})}^{a_{2}^{j}} \times \dots \times {(e^{i γ_{M} ξ_{α}})}^{a_{M}^{j}}, α = 1, 2,

or, equivalently,

b_{α}^{i γ_{j}} = {(b_{α}^{i k_{1}})}^{a_{1}^{j}} {(b_{α}^{i k_{2}})}^{a_{2}^{j}} \times \dots \times {(b_{α}^{i k_{M}})}^{a_{M}^{j}} .

(48)

Consider the multiplicative subgroup Γ of

{(C^{*})}^{2}

generated by

(b_{1}^{i k_{m}}, b_{2}^{i k_{m}}), m = 1, 2, \dots, M,

with the multiplication carried out coordinatewise. Then, the points

(b_{1}^{i k_{j}}, b_{2}^{i k_{j}})

belong to the division group

\bar{Γ}

of Γ, that is,

\bar{Γ} = \{z \in {(C^{*})}^{2} : z^{m} \in Γ for some m \geq 1\} .

In accordance with Lang’s conjecture,¹² the intersection between any algebraic subvariety and the division group for a finitely generated subgroup is along finitely many subtori. The following theorem is proven in Ref. 13:

Theorem

(Ref. 13). Assume that

Γ is a finitely generated subgroup of the multiplicative group of the complex torus ${(C^{*})}^{2},$
$\bar{Γ}$ is the division group of Γ, and
$V \subset {(C^{*})}^{2}$ is an algebraic subvariety given by the zero set of Laurent polynomials.

Then, the intersection of V and

\bar{Γ}

belongs to the union of finitely many translates of certain subtori T₁, …, T_ν contained in V,

V \cap \bar{Γ} = V \cap (T_{1} \cup T_{2} \cup \dots \cup T_{ν}) .

(49)

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 T_j 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:

Theorem 2.

For $ξ_{1} / ξ_{2} \notin Q$ , the Fourier quasicrystal measure μ corresponding to P in (41) satisfies as follows:

a_λ = 1 for λ ∈ Λ, that is, μ is a positive “idempotent.”
$\dim_{Q} Λ = \infty, \dim_{Q} S = 2,$ in particular, μ is not a generalized Dirac comb.
Λ meets any arithmetic progression in $R$ 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 $R$ ), while S is not a Delone set.
$| \hat{μ} |$ is not translation bounded.

Proof.

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 $T^{2}$ 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, a_λ = 1 and μ is a positive idempotent discrete measure.
Since $ξ_{1} / ξ_{2} \notin Q$ ⁠, Lemma 1 implies that $\dim_{Q} Λ_{P} = \infty$ ⁠; 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 of $\hat{μ}$ —belongs to
$L_{+} \cup - L_{+} \cup {0},$
and 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 $(b_{1}^{γ^{*}}, b_{2}^{γ^{*}}) .$ 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 $ξ_{1} / ξ_{2} \notin Q .$ 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 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 γ_j 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 $[n \frac{2 π}{\frac{ξ_{1}}{2} + ξ_{2}}, (n + 1) \frac{2 π}{\frac{ξ_{1}}{2} + ξ_{2}}]$ 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 S_P is not a Delone set, since the measure μ would be a generalized Dirac comb.⁶
Similarly, $| \hat{μ} |$ is not translation bounded, since 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 $| \hat{μ} |$ 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;
The question on p. 3158 of Ref. 2 and part 2 of question 11.2 in Ref. 3:
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 analysis^28–31 to show that general crystalline μ constructed in Sec. II using a stable pair P, Q with parameters b₁, …, b_n, satisfies the following:

Theorem 3.

For such a μ, we have that

Λ = L₁ ⊔ L₂ ⊔ ⋯ ⊔ L_ν ⊔ N, with L₁, …, L_ν full arithmetic progressions and N if not empty is infinite dimensional over $Q$ (the union ⊔ means counted with multiplicities).
a_λ take values in a finite set of positive integers; μ is a positive Fourier quasicrystal.
$\dim_{Q} S = \dim_{Q} \{ξ_{1}, \dots, ξ_{n}\} .$
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 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 $Q .$

ACKNOWLEDGMENTS

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.

REFERENCES

1.

P.

Kurasov

and

P.

Sarnak

, “

The additive structure of the spectrum of a Laplacian on a metric graph

” (unpublished) (

2020

).

Google Scholar

2.

Y. F.

Meyer

, “

Measures with locally finite support and spectrum

,”

Proc. Natl. Acad. Sci. U. S. A.

113

(

12

),

3152

–

3158

(

2016

).

https://doi.org/10.1073/pnas.1600685113

Google Scholar

Crossref

PubMed

3.

N.

Lev

and

A.

Olevskii

, “

Fourier quasicrystals and discreteness of the diffraction spectrum

,”

Adv. Math.

315

,

1

–

26

(

2017

).

https://doi.org/10.1016/j.aim.2017.05.015

Google Scholar

Crossref

4.

A. P.

Guinand

, “

Concordance and the harmonic analysis of sequences

,”

Acta Math.

101

,

235

–

271

(

1959

).

https://doi.org/10.1007/BF02559556

Google Scholar

Crossref

5.

A.

Córdoba

, “

La formule sommatoire de Poisson

,”

C. R. Acad. Sci. Paris Sér. I Math.

306

(

8

),

373

–

376

(

1988

) (in French, with English summary).

Google Scholar

6.

N.

Lev

and

A.

Olevskii

, “

Quasicrystals and Poisson’s summation formula

,”

Invent. Math.

200

(

2

),

585

–

606

(

2015

).

https://doi.org/10.1007/s00222-014-0542-z

Google Scholar

Crossref

7.

Y.

Meyer

,

Nombres de Pisot, Nombres de Salem et Analyse Harmonique: Cours Peccot donné au Collège de France en Avril-mai 1969

, Lecture Notes in Mathematics Vol. 117 (

Springer-Verlag

,

Berlin, NY

,

1970

) (in French).

Google Scholar

Crossref

8.

M. N.

Kolountzakis

, “

Fourier pairs of discrete support with little structure

,”

J. Fourier Anal. Appl.

22

(

1

),

1

–

5

(

2016

).

https://doi.org/10.1007/s00041-015-9416-z

Google Scholar

Crossref

9.

N.

Lev

and

A.

Olevskii

, “

Quasicrystals with discrete support and spectrum

,”

Rev. Mat. Iberoam.

32

(

4

),

1341

–

1352

(

2016

).

https://doi.org/10.4171/RMI/920

Google Scholar

Crossref

10.

D.

Radchenko

and

M.

Viazovska

, “

Fourier interpolation on the real line

,”

Publ. Math. Inst. Hautes Études Sci.

129

,

51

–

81

(

2019

).

https://doi.org/10.1007/s10240-018-0101-z

Google Scholar

Crossref

11.

F. J.

Dyson

,

Birds and Frogs

, Notices of the American Mathematical Society Vol. 56 (

American Mathematical Society

,

2009

), Issue 2, pp.

212

–

223

.

Google Scholar

12.

S.

Lang

, “

Report on diophantine approximations

,”

Bull. Soc. Math. France

93

,

177

–

192

(

1965

).

https://doi.org/10.24033/bsmf.1621

Google Scholar

Crossref

13.

P.

Liardet

, “

Sur une conjecture de Serge Lang

,” in

Journées Arithmétiques de Bordeaux

, Society for Mathematics France, Paris, 1975 (

Conference, University Bordeaux

,

Bordeaux

,

1974

), pp.

187

–

210

, astérisque, Nos. 24-25 (in French).

Google Scholar

14.

J.

Borcea

and

P.

Brändén

, “

The Lee-Yang and Pólya-Schur programs. I. Linear operators preserving stability

,”

Invent. Math.

177

(

3

),

541

–

569

(

2009

).

https://doi.org/10.1007/s00222-009-0189-3.MR2534100

Google Scholar

15.

D. G.

Wagner

, “

Multivariate stable polynomials: Theory and applications

,”

Bull. Am. Math. Soc.

48

(

1

),

53

–

84

(

2011

).

https://doi.org/10.1090/S0273-0979-2010-01321-5

Google Scholar

Crossref

16.

F.

Barra

and

P.

Gaspard

, “

On the level spacing distribution in quantum graphs

,”

J. Stat. Phys.

101

(

1-2

),

283

–

319

(

2000

), special issue on: Dedicated to Grégoire Nicolis on the Occasion of His Sixtieth Birthday (Brussels, 1999).

https://doi.org/10.1023/a:1026495012522

Google Scholar

Crossref

17.

Y.

Colin de Verdière

, “

Semi-classical measures on quantum graphs and the Gaußmap of the determinant manifold

,”

Ann. Henri Poincaré

16

(

2

),

347

–

364

(

2015

).

https://doi.org/10.1007/s00023-014-0326-4

Google Scholar

Crossref

18.

B.

Gutkin

and

U.

Smilansky

, “

Can one hear the shape of a graph?

,”

J. Phys. A: Math. Gen.

34

(

31

),

6061

–

6068

(

2001

).

https://doi.org/10.1088/0305-4470/34/31/301

Google Scholar

Crossref

19.

T.

Kottos

and

U.

Smilansky

, “

Periodic orbit theory and spectral statistics for quantum graphs

,”

Ann. Phys.

274

(

1

),

76

–

124

(

1999

).

https://doi.org/10.1006/aphy.1999.5904

Google Scholar

Crossref

20.

P.

Kurasov

, “

Schrödinger operators on graphs and geometry. I. Essentially bounded potentials

,”

J. Funct. Anal.

254

(

4

),

934

–

953

(

2008

).

https://doi.org/10.1016/j.jfa.2007.11.007

Google Scholar

Crossref

21.

P.

Kurasov

, “

Graph Laplacians and topology

,”

Ark. Mat.

46

(

1

),

95

–

111

(

2008

).

https://doi.org/10.1007/s11512-007-0059-4

Google Scholar

Crossref

22.

D.

Ruelle

,

Thermodynamic Formalism: The mathematical Structures of Classical Equilibrium Statistical Mechanics

, Encyclopedia of Mathematics and its Applications Vol. 5 (

Addison-Wesley Publishing Co.

,

Reading, MA

,

1978

), with a foreword by Giovanni Gallavotti and Gian-Carlo Rota.

Google Scholar

23.

T. D.

Lee

and

C. N.

Yang

, “

Statistical theory of equations of state and phase transitions. II. Lattice gas and Ising model

,”

Phys. Rev.

87

(

2

),

410

–

419

(

1952

).

https://doi.org/10.1103/physrev.87.410

Google Scholar

Crossref

24.

S. Yu.

Favorov

, “

Large Fourier quasicrystals and Wiener’s theorem

,”

J. Fourier Anal. Appl.

25

(

2

),

377

–

392

(

2019

).

https://doi.org/10.1007/s00041-017-9576-0

Google Scholar

Crossref

25.

C.

Lech

, “

A note on recurring series

,”

Ark. Mat.

2

,

417

–

421

(

1953

).

https://doi.org/10.1007/BF02590997

Google Scholar

Crossref

26.

Y. F.

Meyer

, “

Global and local estimates on trigonometric sums

,”

Trans. R. Norw. Soc. Sci. Lett.

2

,

1

–

25

(

2018

).

Google Scholar

27.

J. C.

Lagarias

, “

Mathematical quasicrystals and the problem of diffraction

,” in

Directions in Mathematical Quasicrystals

, CRM Monograph Series Vol. 13 (

American Mathematical Society

,

Providence, RI

,

2000

), pp.

61

–

93

.

Google Scholar

28.

J.-H.

Evertse

, “

Points on subvarieties of tori

,” in

A Panorama of Number Theory or the View from Baker’s Garden

(

Zürich; Cambridge University Press

,

Cambridge

,

1999; 2002

), pp.

214

–

230

.

Google Scholar

Crossref

29.

J.-H.

Evertse

,

H. P.

Schlickewei

, and

W. M.

Schmidt

, “

Linear equations in variables which lie in a multiplicative group

,”

Ann. Math.

155

(

3

),

807

–

836

(

2002

).

https://doi.org/10.2307/3062133

Google Scholar

Crossref

30.

M.

Laurent

, “

Équations diophantiennes exponentielles

,”

Invent. Math.

78

(

2

),

299

–

327

(

1984

).

https://doi.org/10.1007/BF01388597

Google Scholar

Crossref

31.

W. M.

Schmidt

, “

The zero multiplicity of linear recurrence sequences

,”

Acta Math.

182

(

2

),

243

–

282

(

1999

).

https://doi.org/10.1007/BF02392575

Google Scholar

Crossref

2020

Author(s)

Stable polynomials and crystalline measures

I. INTRODUCTION

II. SUMMATION FORMULA

A. Stable polynomials

B. Notations

C. Dirichlet series

D. Logarithmic derivatives

E. Logarithmic derivative as a distribution

F. Summation formula

G. Remarks

III. THE FIRST NON-TRIVIAL EXAMPLE

ACKNOWLEDGMENTS

REFERENCES

Citing articles via

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

Stable polynomials and crystalline measures

I. INTRODUCTION

II. SUMMATION FORMULA

A. Stable polynomials

B. Notations

C. Dirichlet series

D. Logarithmic derivatives

E. Logarithmic derivative as a distribution

F. Summation formula

G. Remarks

III. THE FIRST NON-TRIVIAL EXAMPLE

ACKNOWLEDGMENTS

REFERENCES

Citing articles via

Resources

Explore

pubs.aip.org

Connect with AIP Publishing

This Feature Is Available To Subscribers Only