An example concerning Fourier analytic criteria for translational tiling

It is well-known that the functions $f \in L^1(\mathbb{R}^d)$ whose translates along a lattice $\Lambda$ form a tiling, can be completely characterized in terms of the zero set of their Fourier transform. We construct an example of a discrete set $\Lambda \subset \mathbb{R}$ (a small perturbation of the integers) for which no characterization of this kind is possible: there are two functions $f, g \in L^1(\mathbb{R})$ whose Fourier transforms have the same set of zeros, but such that $f + \Lambda$ is a tiling while $g + \Lambda$ is not.

1. Introduction 1.1. Let f be a function in L 1 (R) and let Λ ⊂ R be a discrete set. We say that f tiles R at level w with the translation set Λ, or that f + Λ is a tiling of R at level w (where w is a constant), if λ∈Λ f (x − λ) = w a.e.
In the same way one can define tiling of R d by translates of a function f ∈ L 1 (R d ).
For example, if f = 1 Ω is the indicator function of a set Ω, and f + Λ is a tiling at level 1, then this means that the translated copies Ω + λ, λ ∈ Λ, fill the whole space without overlaps up to measure zero. To the contrary, for tiling by a general real or complex-valued function f , the translated copies may have overlapping supports.
Tilings by translates of a function have been studied by several authors, see, in particular, [LM91], [KL96], [Kol04], [KL16], [Liu18], [KL21]. The necessity of the condition for tiling in the last example can be generalized as follows. For a discrete set Λ ⊂ R we consider the measure (1.4) We will assume that Λ has bounded density, which means that sup x∈R #(Λ ∩ [x, x + 1)) < +∞. (1.5) In particular (1.5) implies that the measure δ Λ is a temperate distribution on R, so it has a well-defined Fourier transform δ Λ in the distributional sense.
Theorem 1.1 ( [KL16]). Let f ∈ L 1 (R), and Λ ⊂ R be a discrete set of bounded density. If f + Λ is a tiling at some level w, then A similar result is true also in R d . In the earlier works [KL96], [Kol00a], [Kol00b] this result was proved under various extra assumptions.
If Λ is a lattice, then δ Λ = det(Λ) −1 · δ Λ * by the Poisson summation formula. This implies that supp( δ Λ ) = Λ * . Hence in this case the condition (1.6) is not only necessary, but also sufficient, for f + Λ to be a tiling at some level w.
However for a general discrete set Λ of bounded density, the sufficiency of the condition (1.6) for tiling has remained an open problem. In this paper, we settle this problem in the negative. Our main result is the following: Theorem 1.2. There is a discrete set Λ ⊂ R of bounded density (a small perturbation of the integers) with the following property: given any real scalar w there are two realvalued functions f, g ∈ L 1 (R) whose Fourier transforms have the same set of zeros, but such that f + Λ is a tiling at level w while g + Λ is not a tiling at any level.
Moreover, we will show that if the given scalar w is positive, then the functions f, g can be chosen positive as well.
It follows that the necessary condition (1.6) is generally not sufficient for tiling: Corollary 1.3. There exist a set Λ ⊂ R of bounded density and a positive function f ∈ L 1 (R), such that (1.6) is satisfied however f + Λ is not a tiling at any level.
But even stronger, Theorem 1.2 shows that also no other condition can be given in terms of the Fourier zero set Z( f ) that would characterize the functions f ∈ L 1 (R) such that f + Λ is a tiling, even under the extra assumption that f is positive.
Our approach is based on the relation of the problem to Malliavin's non-spectral synthesis example [Mal59c]. The proof involves the implicit function method due to Kargaev [Kar82], who proved the existence of a set Ω ⊂ R of finite measure such that the Fourier transform of its indicator function vanishes on some interval.

Preliminaries. Notation.
In this section we recall some preliminary background and fix notation that will be used later on. For further details we refer the reader to [Kah70].
The closed support of a Schwartz distribution S, or a function φ, on the real line R or on the circle T = R/Z, is denoted by supp(S) or supp(φ) respectively.
If S is a Schwartz distribution on T, its Fourier coefficients S(n) are defined by The action of S on a function φ ∈ C ∞ (T) is denoted by S, φ . We have (2.1) Let A(T) be the Wiener space of continuous functions φ on T whose Fourier series converges absolutely. It is a Banach space endowed with the norm A distribution S on T is called a pseudomeasure if S can be extended to a continuous linear functional on A(T). This is the case if and only if the Fourier coefficients S(n) are bounded. The space P M(T) of all pseudomeasures is a Banach space with the norm The duality between the spaces A(T) and P M(T) is given by which is consistent with (2.1).
In a similar way, we will denote by A(R) the space of Fourier transforms of functions in L 1 (R), that is, φ ∈ A(R) if and only if The Banach space dual to A(R) is then the space P M(R) of temperate distributions S on R whose Fourier transform S is in L ∞ (R). The space P M(R) is normed as and the duality between the spaces A(R) and P M(R) is given by The elements of the space P M(R) are called pseudomeasures on R.
The product φψ of two functions φ, ψ ∈ A (on either T or R) is also in A, and If S ∈ P M and φ ∈ A, then the product Sφ is a pseudomeasure defined by Sφ, ψ = S, φψ , ψ ∈ A, and we have If S ∈ P M, φ ∈ A and if φ vanishes in a neighborhood of supp(S), then Sφ = 0. This is obvious from the definition of supp(S) if φ is a smooth function of compact support, while for a general φ ∈ A this follows by approximation.
If S is a Schwartz distribution on R supported on a compact interval I = [a, b], then its Fourier transform S is an infinitely smooth function on R given by (In fact, S is the restriction to R of an entire function of exponential type).
If S is a distribution on R supported on an interval I of length |I| < 1, then S may be considered also as a distribution on T, and in this case we have S ∈ P M(T) if and only if S ∈ P M(R). If, in addition, φ is a function on R such that supp(φ) ⊂ I, then φ ∈ A(T) if and only if φ ∈ A(R), and the action S, φ then has the same value with respect to either definition (2.2) or (2.3).

Malliavin's non-spectral synthesis phenomenon
3.1. The spectral synthesis problem, posed by Beurling, asks the following: Let V be a closed, linear subspace of the space ℓ ∞ (Z) endowed with the weak* topology (as the dual of ℓ 1 ). We say that V is translation-invariant if whenever a sequence {c(n)} belongs to V , then so do all of the translates of {c(n)}. Define the spectrum σ(V ) of a translation-invariant subspace V to be the (closed) set of points t ∈ T such that the sequence e t := {exp(2πint)} is in V . Is it true that V is generated by the exponentials e t , t ∈ σ(V ), i.e. is V the weak* closure of the linear span of these exponentials?
There are also other, equivalent formulations of the spectral synthesis problem, see [KS94,Chapter IX]. One of them is the following: Let S ∈ P M(T), φ ∈ A(T), and assume that φ vanishes on supp(S). Does it follows that S, φ = 0?
The answer to the last question is affirmative if φ is smooth, or, more generally, if φ ∈ A(T) ∩ Lip( 1 2 ). This result is due to Beurling and Pollard, see e.g. [Kah70, Chapter V, Section 5]. However, it was proved by Malliavin that in the general case, the question admits a negative answer: Theorem 3.1 (Malliavin [Mal59a], [Mal59b]). There exist a pseudomeasure S ∈ P M(T) and a function φ ∈ A(T) such that φ vanishes on supp(S), but S, φ = 0.
The spectral synthesis problem can be posed more generally in any locally compact abelian group G (where the case discussed above corresponds to the group G = Z). For compact groups the problem admits a positive answer; while Malliavin showed [Mal59c] that the answer is negative for all non-compact groups G. 3.2. Let S ∈ P M(T) and φ ∈ A(T) be given by Malliavin's theorem (Theorem 3.1), that is, φ vanishes on supp(S) while S, φ = 0. Since φ does not vanish everywhere on the circle T, there is an open interval I of length |I| < 1 such that supp(S) ⊂ I. Hence we may regard S also as a distribution on R, and we have S ∈ P M(R). By multiplying φ on a smooth function supported on I and which is equal to 1 in a neighborhood of supp(S), we may assume that supp(φ) ⊂ I as well, and consequently φ ∈ A(R).
Furthermore, by applying a linear change of variable to S and φ, we may actually suppose that I is an arbitrary open interval on R. We shall take I = (a, b) where a, b are any two numbers satisfying 0 < a < b < 1 2 . For each r > 0 we now define a distribution T r ∈ P M(R) by where S(t) := S(−t). 1 We will prove the following result: Theorem 3.2. Given any ε > 0 there exists a real sequence Λ = {λ n }, n ∈ Z, satisfying |λ n − n| ε for all n, such that for some r > 0 we have The proof of this theorem will be given in the next section. Our goal in the present section is to complete the proof of Theorem 1.2 based on this result. We will show that Λ has the property from the statement of the theorem: given any real scalar w there are two real-valued functions f, g ∈ L 1 (R) whose Fourier transforms have the same set of zeros, but such that f + Λ is a tiling at level w while g + Λ is not a tiling at any level. Moreover, if the given scalar w is positive, then the functions f, g can be chosen positive as well.
3.3. Since the set Λ has bounded density, for any h ∈ L 1 (R) the convolution h * δ Λ is a locally integrable function satisfying Proof. The assertion means that for any Schwartz function β we have Let χ be a Schwartz function whose Fourier transform χ is nonnegative, has compact support, χ(t)dt = 1, and for each ε > 0 let χ ε (x) := χ(εx). Let q ε := ( h · β) * χ ε , then q ε is an infinitely smooth function with compact support. As ε → 0, the function q ε remains supported on a certain closed interval J contained in (−b, b), and q ε converges to h · β in the space A(R). The assumption that (3.4) The function β is the Fourier transform of some function α in the Schwartz class. Let p ε := (h * α) · χ ε , then p ε is a smooth function in L 1 (R) and we have p ε = q ε . Since q ε belongs to the Schwartz space, the same is true for p ε , and it follows that (3.5) Now we need the following: We observe that |χ ε (−λ)| 1 and χ ε (−λ) → 1 as ε → 0 for each λ. Hence the claim allows us to apply the dominated convergence theorem to the sum (3.5), which yields (3.7) The claim also allows us to exchange the sum and integral in (3.7), and it follows that Comparing (3.4) and (3.8), we see that (3.3) holds.
It remains to prove the claim. Indeed, we have The inner sum on the right hand side of (3.9) is a bounded function of x, since α is a Schwartz function and Λ has bounded density, while h is a function in L 1 (R). Hence the integral in (3.9) converges, and this completes the proof of the lemma.
Given a real scalar w we define two functions f, g ∈ L 1 (R) by the conditions (3.10) (3.11) then f, g are real-valued and their Fourier transforms have the same set of zeros.
By Lemma 3.3 the Fourier transform of f * δ Λ is the pseudomeasure where the first equality is due to (3.1) and (3.10), while the second equality is true since ψ is smooth and vanishes on supp(S), hence Sψ = 0. We conclude that f * δ Λ = w a.e., which means that f + Λ is a tiling at level w.
However in this case, Sφ is not the zero distribution, since Sφ, 1 = S, φ = 0. This shows that the Fourier transform of g * δ Λ is not a scalar multiple of δ 0 , and it follows that g + Λ is not a tiling at any level.
3.5. The above construction yields real-valued functions f and g, but these two functions need not be positive. We will now show that if the given scalar w is positive, then the construction can be modified so as to yield everywhere positive functions f, g.
In what follows, φ and ψ continue to denote the same two functions as above.
(3.13) Indeed, we may apply the same procedure from Step 1 also to the function ψ, and then define {c(k)} to be the maximum of the two sequences obtained from both steps.
Step 4 : Now suppose that we are given a positive scalar w. We then define the two functions f, g by the conditions (3.17) We observe that by the definition of the function τ we have and in particular τ (0) is nonzero. Then f, g are in L 1 (R), their Fourier transforms have the same set of zeros, and by the same argument as before one can verify that f + Λ is a tiling at level w, while g + Λ is not a tiling at any level.
Finally we check that f and g are everywhere positive functions. Indeed, we have and by (3.12), (3.13), (3.14) and (3.15) it follows that f (x), g(x) > 0 for every x ∈ R.
This completes the proof of Theorem 1.2 based on Theorem 3.2.
It remains to prove Theorem 3.2. This will be done in the next section. The question was answered in the affirmative by Kargaev [Kar82]. The solution was based on an innovative application of the infinite-dimensional implicit function theorem, which established the existence of a set of the form Ω = n∈Z [n + α n , n + β n ], where {α n }, {β n } are two real sequences in ℓ 1 , that has the above mentioned property.
The approach was later used in [KL16] in order to prove the existence of non-periodic tilings of R by translates of a function f . In that paper, a self-contained presentation of the method was given in a simplified form, that does not invoke the infinite-dimensional implicit function theorem.
In this section, we use an adapted version of Kargaev's method in order to prove Theorem 3.2 (and more, in fact). The presentation below generally follows the lines of [KL16, Sections 2, 3], but the proof also requires some additional arguments. 4.2. Let {α n }, n ∈ Z, be a bounded sequence of real numbers. To such a sequence we associate a function F on the real line, defined by where F n is the function 1 [n,n+αn] if α n 0, or −1 [n+αn,n] if α n < 0.
Since the sequence {α n } is bounded, the series (4.1) is easily seen to converge in the space of temperate distributions to a bounded function F on R. In particular, F is a temperate distribution.
Theorem 4.1. Given two numbers b ∈ (0, 1 2 ) and ε > 0, there is δ > 0 with the following property: Let S be a Schwartz distribution on R satisfying Then there is a bounded, real sequence α = {α n }, n ∈ Z, such that α ∞ ε and where F is the function defined by (4.1).
The proof of Theorem 4.1 is given below. It is divided into a series of lemmas.
Let I := [− 1 2 , 1 2 ]. Denote by ψ(k) the k'th Fourier coefficient of a function ψ on I: The following lemma is inspired by [KV92, Lemma 2.2]. Proof. First suppose that k = 0. We have |ϕ(t)| C|t| for t ∈ I, hence | ψ s (0)| C|s| I |tΦ(t)|dt = C|s|. (4.6) Next we assume that k = 0. We integrate by parts m times and use the fact that the function ψ s vanishes in a neighborhood of the points ± 1 2 . This yields By the product rule for the m'th derivative we have (4.8) Combining (4.7) and (4.8) yields the estimate Since the derivatives ϕ ′ , ϕ ′′ , . . . , ϕ (m) are bounded on I, each one of the terms in the sum corresponding to j = 1, 2, . . . , m is bounded by C|s|, while the term corresponding to j = 0 can be estimated using |ϕ(t)| C|t|, t ∈ I, which again yields C|s|.
(4.9) Proof. We may suppose that u < v. We observe that where we define ψ s (t) := ϕ ′ (st)Φ(t). Hence converges unconditionally in the distributional sense to T in the open interval (− 1 2 , 1 2 ). This follows from the unconditional convergence of the series (4.14) to T considered as a distribution on the circle T. 4.5. Let X be the space of all bounded sequences of real numbers α = {α n }, n ∈ Z, endowed with the norm α X := sup n∈Z |α n | that makes X into a real Banach space.
Let Y be the space of distributions T supported on [−l, l] whose Fourier coefficients T (k), k ∈ Z, are real and bounded. If we endow Y with the norm then also Y is a real Banach space, which can be viewed as a closed subspace of P M(T).
We observe that a distribution T supported on [−l, l] has real Fourier coefficients (that is, T (k) ∈ R for every k ∈ Z) if and only if T (−t) = T (t).
Lemma 4.5. Let {T n }, n ∈ Z, be a sequence of elements of Y . Assume that there is a sequence γ ∈ X such that | T n (k)| |γ n | 1 + |k − n| 2 (4.16) for every n and k in Z. Then the series converges unconditionally in the distributional sense to an element T ∈ Y satisfying where K is an absolute constant.
Proof. Indeed, the condition (4.16) implies that for any φ ∈ A(T) we have This shows that the series (4.17) converges unconditionally in the weak* topology of the space P M(T) (the dual of A(T)) to an element T ∈ Y satisfying (4.18). 4.6. Let α = {α n }, n ∈ Z, be a sequence in X such that α X 1. Define Let T n be the n'th term of the series (4.19). We observe that T n ∈ Y . If we apply Lemma 4.2 to the function ϕ(t) := (e 2πit − 1 − 2πit)/(2πit) with s = α n and m = 2, then it follows from the lemma that condition (4.16) is satisfied with γ n := Cα 2 n , where C > 0 does not depend on α, k or n. Hence by Lemma 4.5 the series (4.19) converges in the distributional sense to an element of the space Y , and we have Rα Y C α 2 X , α ∈ X, α X 1, (4.20) where the constant C does not depend on α.
We note that the mapping R defined by (4.19) is nonlinear. 4.7. For each r > 0 let U r denote the closed ball of radius r around the origin in X: Lemma 4.6. Given any ρ > 0 there is 0 < r < 1 such that In particular, if r is small enough then R is a contractive (nonlinear) mapping on U r .
Proof. Let α, β ∈ U r (0 < r < 1). Then using (4.19) we have (4.23) Let T n be the n'th element of the series (4.23). We apply Lemma 4.3 to the function ϕ(t) := (e 2πit − 2πit)/(2πi) with u = α n , v = β n and m = 2. The lemma implies that the condition (4.16) is satisfied with γ n := Cr · (β n − α n ), where the constant C does not depend on r, α, β, k or n. It therefore follows from Lemma 4.5 that we have the estimate Rβ − Rα Y Cr β − α X where C is a constant not depending on r, α or β. Hence it suffices to choose r small enough so that Cr ρ.
Hence if we choose δ such that C(1 + ε) 2 δ ε then we obtain and it follows that H(B) ⊂ B.
It also follows from Lemma 4.6 that if δ is small enough, then H is a contractive mapping from the closed set B into itself. Indeed, let T 1 , T 2 ∈ B, then we have where 0 < ρ < 1. Then the Banach fixed point theorem implies that H has a (unique) fixed point T ∈ B, which yields the desired solution.
4.9. Proof of Theorem 4.1. Let S be a Schwartz distribution satisfying (4.2). Then S ∈ Y and S Y δ. Define S 1 (t) := S(−t), then also S 1 is a distribution in Y and we have S 1 Y = S Y . By Lemma 4.7, if δ is small enough then there is an element Let α ∈ X be the sequence defined by α n = T (n), n ∈ Z, then α X ε provided that δ is small enough. Let F be the function given by (4.1) that is associated to this sequence α = {α n }. We have in the sense of distributions, and The first sum converges to T in (−b, b) according to Lemma 4.4; while the second sum converges to Rα in (−b, b), which is due to (4.19) and the fact that This means that F = S in (−b, b) and thus Theorem 4.1 is proved.
4.10. The theorem just proved will now be used to deduce the following one: Theorem 4.8. Given two numbers a, b such that 0 < a < b < 1 2 , and given ε > 0, there is δ > 0 with the following property: Let S be a distribution on R satisfying (4.24) Then there is a real sequence Λ = {λ n }, n ∈ Z, such that |λ n − n| ε for all n, and Proof. We choose an infinitely smooth function Ψ such that Ψ(−t) = Ψ(t) for all t ∈ R, (a, b), and Ψ(t) = 0 for t ∈ R \ (−l, l).
Let S be a distribution satisfying (4.24), then S ∈ Y and S Y δ. Define a new distribution S 1 := S · Ψ, then also S 1 ∈ Y . We have By Theorem 4.1, if δ is small enough then there is a sequence α ∈ X, α X ε, such that the function F defined by (4.1) satisfies F = S 1 in (−b, b). It follows that the distributional derivative F ′ of the function F satisfies which is true since 2πitΨ(t) = −1 in a neighborhood of supp(S).
Let Λ = {λ n }, n ∈ Z, be defined by λ n := n + α n . Then we have |λ n − n| ε for all n. It follows from the definition (4.1) of F that due to (4.26). The proof of Theorem 4.8 is thus concluded. 4.11. Finally, we observe that Theorem 3.2 follows from Theorem 4.8. Indeed, if S is a pseudomeasure on R such that supp(S) ⊂ (a, b), then the distribution r(S + S) satisfies the conditions (4.24) if r > 0 is sufficiently small. Hence Theorem 4.8 yields a sequence Λ = {λ n } with the properties as in the statement of Theorem 3.2.

Addendum: A problem of Kolountzakis
The following question was posed to us by Kolountzakis: Does there exist a real sequence Λ = {λ n }, n ∈ Z, satisfying where A, B > 0 are constants, such that f + Λ is a tiling for some nonzero f ∈ L 1 (R), but there is no nonnegative f with this property?
The answer turns out to depend on the level of the tiling. Suppose first that there is a tiling f + Λ at some nonzero level w. Then f must have nonzero integral, see [KL96, Lemma 2.3(i)]. In turn this implies [KL16, Section 4] that δ Λ = c · δ 0 in some neighborhood (−η, η) of the origin, where c is a nonzero, positive scalar. It follows that f + Λ is a tiling whenever f is a Schwartz function with supp( f ) ⊂ (−η, η). In particular, there exist tilings f + Λ at level one with f nonnegative.
To the contrary, we will construct an example showing that the same is not true if Λ is only assumed to admit a tiling at level zero. We will prove the following result: Theorem 5.1. There is a real sequence Λ = {λ n }, n ∈ Z, satisfying (5.1) for which there exist tilings f + Λ with nonzero f ∈ L 1 (R), but any such a tiling is necessarily a tiling at level zero. In particular Λ cannot tile with any nonnegative (nonzero) f . Proof. Let a, b be two numbers such that 0 < a < b < 1 2 . Let ψ be a smooth even function, ψ(t) > 0 on (−a, a), and ψ(t) = 0 outside (−a, a). By Theorem 4.1, given any ε > 0 there is a real sequence α = {α n }, n ∈ Z, satisfying |α n | ε for all n, and such that F (t) = rψ(t) in (−b, b) for some r > 0, where F is the function defined by (4.1).
Let the sequence Λ = {λ n }, n ∈ Z, be defined by λ n := n + α n . Then This can be shown in the same way as done in the proof of Theorem 4.8 above.
In particular we have δ Λ = 0 in the open set G := (−b, −a) ∪ (a, b), so there exist nonzero real-valued Schwartz functions f such that f + Λ is a tiling (at level zero). It suffices to choose f such that supp( f ) is contained in G.
On the other hand, suppose that there is f ∈ L 1 (R) such that f + Λ is a tiling at some nonzero level w. Then, as before, this implies that δ Λ = c · δ 0 in some interval (−η, η), where c is a nonzero (positive) scalar. This contradicts (5.2), hence no such f exists. In particular, if f is nonnegative and f + Λ is a tiling, then the tiling level must be zero and f vanishes a.e. 6. Remarks 6.1. Let Λ ⊂ R be a discrete set of bounded density. If the temperate distribution δ Λ is a measure on R, then condition (1.6) is not only necessary, but also sufficient, for a function f ∈ L 1 (R) to tile at some level w with the translation set Λ. In this case the tiling level is given by w = c(Λ) f (0), where c(Λ) is the mass that the measure δ Λ assigns to the origin (see [KL21, Theorem 2.2]).
For example, if Λ is a periodic set then δ Λ is a (pure point) measure, and f + Λ is a tiling if and only if (1.6) holds. It follows that the set Λ in Theorem 1.2 is not periodic, nor can it be represented as a finite union of periodic sets.
6.2. If f has fast decay, e.g. |f (x)| = o(|x| −N ) as |x| → +∞ for every N, then f is a smooth function and again the condition (1.6) is both necessary and sufficient for f + Λ to be a tiling at some level w. It follows that the function f in Corollary 1.3 cannot be chosen to have fast decay. 6.3. Theorem 1.2 also holds in R d for every d 1. This can be easily deduced from the one-dimensional result by taking cartesian products. For example, in R 2 one may take F (x, y) = f (x)h(y), G(x, y) = g(x)h(y), where f , g are the functions from Theorem 1.2 and where h ∈ L 1 (R) is such that h + Z is a tiling at level one. Then F , G have the same set of zeros, but F tiles with the translation set Λ × Z while g does not.