Fourier quasicrystals with unit masses

Every set $\Lambda\subset R$ such that the sum of $\delta$-measures sitting at the points of $\Lambda$ is a Fourier quasicrystal, is the zero set of an exponential polynomial with imaginary frequencies.


Introduction
By a crystalline measure one means an atomic measure µ which is a tempered distribution and whose distributional Fourier transform is also an atomic measure, µ = A few constructions of aperiodic FQs are known, see [LO16], [Kol16], [M16].Recently P. Kurasov and P. Sarnak [KS20] found examples of FQs with unit masses, where Λ is an aperiodic uniformly discrete set in R.
Alternative approaches to construction of such measures were suggested by Y. Meyer [M20] and later in [OU20].In the present note we show that the construction in the last paper characterises all FQs of the form (1): Theorem 1 Let µ be an FQ of the form (1). Then there is an exponential polynomial with real simple zeros such that Λ is the zero set of p.
In the opposite direction, the construction in [OU20] shows that for every exponential polynomial p with real simple zeros there is an FQ µ with unit masses whose support Λ is the zero set of p.

Auxiliary Results
We will use the standard form of Fourier transform Let us start with a result which may have intrinsic interest: Proposition 1 Let µ be a positive measure which is a tempered distribution, such that its distributional Fourier transform μ is a measure satisfying which means that |μ| is a tempered distribution.Then there exists C such that Proof.It suffices to prove (3) for every interval (a, b) satisfying b − a ≥ 2. Fix any non-negative Schwartz function g(x) supported by [−1/2, 1/2] and such that Using this inequality and (2), we get On the other hand, clearly, A set Λ is called relatively uniformly discrete if it is a union of finite number of uniformly discrete sets.Proposition 1 implies Corollary 1 Let µ be a measure of the form (1) whose distributional Fourier transform is a measure satisfying (2).Then its support Λ is a relatively uniformly discrete set.
In what follows we assume that µ is an FQ of the form (1). Recall that its Fourier transform μ admits a representation where S is a locally finite set and |μ| satisfies (2).We assume that 0 ∈ Λ (otherwise, we consider the measure µ(d(x − a)), for an appropriate a).
Then the inverse Fourier transform of e w is given by Denote by s 1 ∈ S, s 1 < 0, the greatest negative element of S, were S is the spectrum of μ in (4).Choose any non-negative function H ∈ S(R) which vanishes outside of (−1, 0) and such that R H(t) dt = 1.Also, denote by h the inverse Fourier transform of H.
The series above converges absolutely due to (2).Letting w = i, we get where a s e 2πs .

Now, by Corollary 1 the series converges
Hence, in the formula above we may let ǫ → 0 to get a s e −2πiws .
This proves (5), where α is defined as The series above converges absolutely due to Corollary 1.
Similarly to the proof above, one can establish a variant of (5) for the lower half-plane: where β ∈ C.

Proof of Theorem 1
It follows from Corollary 1 that The inequality above and a theorem of Borel (see [L96], Lecture 4, Theorem 3) imply that ψ is an entire function of order one, i.e. for every ǫ > 0 there is a constant C(ǫ) such that It is clear that we have This and (5) imply the representation ψ(w) = Ce αw e s∈S,s<0 (as/s)e −2πisw , Im w > 0, with some C ∈ C. Set p(w) := ψ(w)e −αw .
Since ψ is of order one, the same is true for p.
Observe that p has simple real zero at the points of Λ.To prove the theorem, we show that p is an exponential polynomial with imaginary frequencies.
This shows that p is bounded from above in any half-plane Imw ≥ r, r > 0.
Similarly, by ( 6 We now apply a variant of the Phragmén-Lindelöf principle (see [L96], Lecture 6, Theorem 2) in the angles {|arg w| ≤ π/4} and {|π−arg w| ≤ π/4} to prove that p is an entire functions of exponential type, i.e.One may check that U is a locally finite set and that the series in (11) converges absolutely for every w ∈ C, Im w > 0.
To prove the theorem, we have to show that the series in the right hand-side of (11) contains only a finite number of terms.

λ∈Λ c λ δ
λ , μ = s∈S a s δ s , c λ , a s ∈ C, where the support Λ and the spectrum S of µ are locally finite sets.If in addition the measures |µ| and |μ| are also tempered, then µ is called a Fourier quasicrystal (FQ).A classical example of an FQ is a Dirac comb µ = n∈Z δ n , which satisfies μ = µ.Here δ x denotes the Dirac measure at point x.