# On the Result of Invariance of the Closure Set of the Real Projections of the Zeros of an Important Class of Exponential Polynomials

Authors: J. M. Sepulcre
Journal: Journal of Function Spaces
 (2016)

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## Abstract

We provide the proof of a practical pointwise characterization of the setRP defined by the closure set of the real projections of the zeros of an exponential polynomial P(z)=∑j=1ncjewjz with real frequencies wj linearly independent over the rationals. As a consequence, we give a complete description of the set RP and prove its invariance with respect to the moduli of the cj′s, which allows us to determine exactly the gaps of RP and the extremes of the critical interval of P(z) by solving inequations with positive real numbers. Finally, we analyse the converse of this result of invariance.


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## Body

## 1. Introduction

Throughout this paper we will consider exponential polynomials of the form(1)Pz=c1ew1z+⋯+cnewnz,cj∈C∖0,n≥2,where {w1,w2,…,wn} is an ordered set of real numbers, called frequencies of P(z), which are linearly independent over the field of the rational numbers. It is known that P(z) is an entire function of order 1 having infinitely many zeros located on a vertical strip (see, e.g., [1, Lemma  2.5]), the critical strip of P(z), bounded by the real numbers aP and bP, where(2)aP≔inf⁡Rez:Pz=0,(3)bP≔sup⁡Rez:Pz=0.It is worth noting that the research devoted to studying the zeros of exponential polynomials appears in the first third of the twentieth century in relation to the development of differential equation theory [2] (see also [3, 4]). In this respect, it is related to the study of the location of the characteristic roots of linear delay differential equations (see, e.g., [5, 6]), and, more so, the zeros of exponential polynomials are the roots of the characteristic equations of the associated functional difference equations which determine the essential spectrum of the solution operators of the associated neutral functional differential equations [7].The study of the zeros of the class of exponential polynomials of type (1) has recently become a topic of increasing interest. In fact, in connection with other areas of study, it allowed to prove that a conjecture on the set of dimensions of fractality associated with a nonlattice fractal string is true in the case of generic nonlattice self-similar strings [8, Theorem  13], which have a clear correspondence with the exponential polynomials of type (1).Given such a functionP(z), the bounds aP and bP allow us to define an interval IP≔aP,bP, called critical interval of P(z), which contains the closure of the set of the real parts of the zeros of P(z), denoted by(4)RP≔Rez:Pz=0¯.In this respect, the density properties of the zeros of several types of exponential polynomials were studied, for example, in [9–13] and more recently in [8, 14–18]. At this point, we must also mention the high sensitivity of the bounds of the critical interval to the frequency variations. Furthermore, the results on the nature of RP have immediate implication to the theory of stability for neutral time delay systems. In fact, from the stability point of view, it is important to study the behaviour of the smallest upper bound of the spectrum [7, 9, 19–24]. It was shown by Avellar and Hale [9] that, in general, the smallest upper bound is not continuous with respect to the delays (i.e., arbitrarily small changes in the delays destabilise the associated neutral functional differential equation). However, the continuity is reached when the delays are rationally independent [9, Theorem  2.2] and, in that case, the set RP is completely characterized by [9, Theorem  3.1, Corollary  3.2].In this way, we will suppose in this paper that the frequencies of our exponential polynomialsP(z) are rationally independent in order to make good use of the accurate knowledge of the nature of the set RP. In fact, concerning exponential polynomials of type (1), Moreno proved in 1973 the following theorem which we quote:(Moreno [12, Main Theorem]) “Assume that1,α1,…,αmare real numbers linearly independent over the rationals. Consider the exponential polynomial(5)φz=∑k=1mAkeαkz,z=σ+it,where theAkare complex numbers. Then a necessary and sufficient condition forφ(z)to have zeros arbitrarily close to any line parallel to the imaginary axis inside the strip(6)I=σ+it:σ0<σ<σ1,-∞<t<∞is that(7)Ajeσαj≤∑k=1,k≠jmAkeσαk,j=1,2,…,mfor anyσwithσ+it∈I.”From this result it follows that an open interval(σ0,σ1) is included in Rφ≔Rez:φ(z)=0¯ if and only if geometrical principle (7) holds on (σ0,σ1). It is worth noticing that Moreno’s result (another version of this theorem which can be found in [8, Theorem  1]) is only applicable to open intervals and hence it does not identify the boundary points of Rφ. This characterization was used in [8] in order to establish a bound for the number of gaps of the sets defined in (4).On the other hand, the topological properties of the setRζn=Rez:ζn(z)=0¯ associated with the partial sums ζnz=1+2-z+⋯+n-z, n≥2, of the Riemann zeta function have been studied from different approaches. For example, an auxiliary function associated with ζn(z) was used in [17, Theorem  9] in order to establish conditions to decide whether a real number is in the set Rζn. This auxiliary function specifically arises from a known result of Avellar and Hale [9, Theorem  3.1] which leads us to analytical criteria in the case of exponential polynomials whose frequencies are not necessarily linearly independent over the rationals.In this paper, by following the ideas of [17] and based on inequalities (7), we will get a practical characterization of the points of the set RP associated with an exponential polynomial P(z) of type (1). Also, in order to prove this pointwise geometrical principle and to obtain other preliminary properties, we will use Kronecker’s theorem which is stated under the following form:(Hardy and Wright [25, p. 382]) “Leta1,a2,…,ambe a linearly independent set of non-null real numbers. For arbitrary real numbersb1,b2,…,bmandT,ε>0, there exist a real numbert>Tand integersn1,n2,…,nmsuch that(8)tak-nk-bk<ε,∀k=1,2,…,m.”In such a way, related to an exponential polynomialP(z) of type (1), in this paper we specifically prove the following:(i)
Fixσ0∈R; the closure of the set Img(P(σ0+it))={s∈C:∃t∈Rsuchthats=P(σ0+it)} coincides with the image set of an auxiliary function, denoted by FP(σ0,x), associated with P(z) (see Proposition 3).(ii)
A real numberσ0∈RP if and only if geometrical principle (35) holds on σ0 (see Theorem 6), which constitutes a practical pointwise characterization of the closure set of the real parts of the zeros of exponential polynomials whose frequencies are linearly independent over the rationals.(iii)
RP is the union of, at most, n-1 disjoint nondegenerate closed intervals (see Theorem 9 and its demonstration which constitutes another shorter proof of [8, Theorem  9]).(iv)
RP is invariant with respect to the moduli of the cj′s (see Corollary 10), which let us exert a control about the number of gaps of RP by means of an adequate selection of the moduli of the coefficients associated with P(z).(v)
If two exponential polynomials with two or three terms are written in their normalized forms and they have the same setRP, then the moduli of their coefficients are pairwise equal (see Proposition 12), which for these cases is a kind of converse assertion to the invariance result formulated in the previous point. This result cannot be extended to exponential polynomials with more than three terms (see Example 13).


## 2. An Auxiliary Function Associated withP(z)

LetP(z) be of form (1) with n=2. Equation c1ew1z+c2ew2z=0 gives us the equality e(w2-w1)z=-c1/c2. Hence the zeros of P(z) for n=2 are given by (9)zk=log⁡c1/c2+iarg-c1/c2+2kπw2-w1,withk∈Z.Therefore, in this case, the set RP, defined in (4), is equal to the point logc1/c2/(w2-w1) and then, from now on, we will suppose that n>2.We next define an auxiliary function, associated with the exponential polynomialsP(z) of form (1), which will be used in this paper in order to find some characterizations of the set RP defined in (4). This function is clearly inspired by [9, Theorem  3.1].Definition 1.
LetP(z)=c1ew1z+⋯+cnewnz be an exponential polynomial of type (1). We define the auxiliary function FP:R×Rn→C associated with P(z) as (10)FPσ,x≔∑m=1ncmewmσexmi,σ∈R,x=x1,x2,…,xn∈Rn.We next introduce the following notation which will be used to show the direct relation between an exponential polynomial of type (1) and the auxiliary function associated with it.Notation 1.
GivenP(z) an exponential polynomial of type (1) and fixing σ0∈R, we consider the following sets of the complex plane: (11)ImgPσ0+it≔s∈C:∃t∈Rsuch  thats=Pσ0+it,ImgFPσ0,x≔s∈C:∃x∈Rnsuch  thats=FPσ0,x.Fixingσ0∈R, we now establish the direct inclusion between the sets above.Lemma 2.
LetP(z) be an exponential polynomial of type (1) and σ0∈R; then (12)ImgPσ0+it⊂ImgFPσ0,x.Proof.
Fixingσ0∈R, consider that P(z)=c1ew1z+⋯+cnewnz of type (1) and s0∈Img(P(σ0+it)), then s0=P(σ0+it0) for some t0∈R; that is,(13)s0=∑m=1ncmewmσ0eiwmt0.On the other hand, from Definition 1, for any x=(x1,x2,…,xn)∈Rn, we have(14)FPσ0,x=∑m=1ncmewmσ0eixm.Therefore, by taking xj=t0wj for j=1,2,…,n, expression (14) becomes (13). Consequently, s0∈Img(FP(σ0,x)) and the lemma is proved.In fact, fixingσ0, by using Kronecker’s theorem, we next prove that Img(FP(σ0,x)) coincides with the closure of the set Img(P(σ0+it)).Proposition 3.
LetP(z) be an exponential polynomial of type (1). Then, for every σ0∈R, it is verified that (15)ImgFPσ0,x=ImgPσ0+it¯.Proof.
Givenn≥2, consider that P(z)=c1ew1z+⋯+cnewnz, with cj∈C∖{0}, j=1,2,…,n, and {w1,w2,…,wn} is an ordered set of real numbers linearly independent over the rationals. Fix σ0∈R. Notice that Img(P(σ0+it))⊂Img(FP(σ0,x)) in virtue of Lemma 2. We are going to demonstrate that Img(P(σ0+it)) is dense in Img(FP(σ0,x)) or, equivalently, given δ>0 and w∈Img(FP(σ0,x)) there exists t1∈R such that (16)w-Pσ0+it1<δ.Indeed, if w∈Img(FP(σ0,x)), then there exists a vector x=(x1,x2,…,xn) such that(17)w=FPσ0,x=∑m=1ncmewmσ0eixm.Fixing T∈R and ε>0, we next apply Kronecker’s Theorem [25, p. 382] with the following choice: (18)aj=wj2π,j=1,2,…,n;bj=xj2π,j=1,2,…,n.Hence there exist some integers N1,N2,…,Nn and t1>T such that (19)t1wj2π-Nj-xj2π<ε,j=1,2,…,n;that is,(20)t1wj=2πNj+xj+ηj,j=1,2,…,nwith |ηj|<2πε, j=1,…,n. Therefore, since (21)Pσ0+it1=∑m=1ncmewmσ0eiwmt1,from (17) and (20), we have (22)w-Pσ0+it1=∑m=1ncmewmσ0eixm-eiwmt1≤∑m=1ncmewmσ0eixm-ei2πNm+xm+ηm=∑m=1ncmewmσ01-eiηmwhich goes to 0 when ε tends to 0 because |ηj|<2πε, for j=1,…,n. Consequently, we conclude that (23)ImgPσ0+it¯=ImgFPσ0,x.


## 3. Characterizations of the Closure Set of the Real Projections of the Zeros

This section is devoted to showing some pointwise characterizations of the setsRP defined in (4). We previously need the next lemma (compare with [14, Lemma  3]).Lemma 4.
LetP(z) be an exponential polynomial of type (1), σ0 a point in aP,bP, and (Tj)j=1,2,… a real sequence such that (24)limj→∞⁡Pσ0+iTj=0.Then σ0∈RP.Proof.
In order to apply [12, Lemma, p. 73], we next prove the following two conditions that joint with the assumption of the existence of a sequence of points {σ0+iTj} such that P(σ0+iTj)→0; let us prove this lemma: (a)
P(z) is bounded on its critical strip SP≔z=σ+it:aP≤σ≤bP.(b)
There exist positive numbersδ and l such that on any segment of length l of the line x=σ0 there is a point σ0+iM such that P(σ0+iM)≥δ.
Givenz=σ+it, we first observe that (25)Pz≤c1ew1z+⋯+cnewnz=c1ew1σ+⋯+cnewnσ≤c1+⋯+cnewuσ,where wuσ=max⁡{wjσ:j=1,…,n}. Hence(26)Pz≤c1+⋯+cnewuAprovided that Rez≤A. Then P(z) is, in particular, bounded on its critical strip SP and we deduce (a). On the other hand, let t0 be a real number such that |P(σ0+it0)|>0 and take δ=|P(σ0+it0)|/2. Then, since P(z) is an almost-periodic function [26, p. 101], there exists a real number l=l(δ) such that every interval of length l on the imaginary axis contains at least one translation number iT, associated with δ, satisfying P(z+iT)-P(z)≤δ for all z∈C. Thus, by taking z=σ0+it0, we have P(σ0+i(t0+T))-P(σ0+it0)≤δ and, according to the choice of δ, it follows that P(σ0+i(t0+T))≥δ. Therefore, P(σ0+iM)≥δ, with M≔t0+T, which proves (b). Consequently, the function P(z) has the properties needed to apply [12, Lemma, p. 73] and thus P(z) has zeros in any strip Sϵ≔z:σ0-ϵ<Rez<σ0+ϵ, for arbitrary ϵ>0. Then σ0∈RP. This completes the proof.The previous lemma provides a sufficient condition to get points ofRP. It allows us to give a characterization of the sets RP by means of anad hoc version of [9, Theorem  3.1], which is obtained through the auxiliary function analysed in the previous section.Theorem 5.
LetP(z) be an exponential polynomial of type (1). Then a real number σ∈RP if and only if there exists some vector x∈Rn such that FP(σ,x)=0, where FP is the auxiliary function associated with P(z) defined in (10).Proof.
For any zeroz=σ+it of P(z) we have(27)0=Pz=∑m=1ncmewmz=∑m=1ncmewmσewmti=FPσ,tw,where w≔(w1,w2,…,wn). Assume that σ∈RP. Then there exists a sequence (sj=σj+itj)j=1,2,… of zeros of P(z) such that σ=limj→∞σj and, from (27), we get that FP(σj,tjw)=0 for each j=1,2,…, which is equivalent to writing(28)0=∑m=1ncmewmσjewmtji,∀j=1,2,….For m=1, since the sequence ew1tjij=1,2,… is on the unit circle, it is bounded, so there exists a subsequence ew1tjh,1ih=1,2,… convergent to a point eθ1i for some θ1∈0,2π. For m=2, from the sequence ew2tjij=1,2,…, we can draw a subsequence ew2tjh,2ih=1,2,… convergent to a point eθ2i for some θ2∈0,2π and so successively for each m∈{1,2,…,n}. Then, by considering (28) for j∈jh,mh=1,2,… and taking the limit when h→∞, we obtain (29)0=∑m=1ncmewmσjh,mewmtjh,mi=∑m=1ncmewmσeθmi=FPσ,θ,where θ≔(θ1,θ2,…,θn).
Conversely, suppose thatFP(σ,x)=0 for some real number σ and a vector x=(x1,x2,…,xn) of Rn. Thus, noticing that the components of the vector (30)12πw=12πw1,…,12πwnare linearly independent over the rationals, given the numbers (1/2π)x1,(1/2π)x2,…,(1/2π)xn; T=1 and ε/2j2π, with ε>0, by applying Kronecker’s theorem [25, p. 382] for each fixed j=1,2,…, there exist Tj>1 and integers Nj,ll=1,2,…,n such that for each j=1,2,…, we have(31)Tj12πwl-12πxl-Nj,l<ε2j2π∀l=1,2,…,n.Multiplying by 2π, (31) becomes (32)Tjwl-xl-2πNj,l<ε2j∀l=1,2,…,n,which implies that (33)limj→∞⁡exl-Tjwli=1,∀l=1,2,…,n.Then we are led to(34)limj→∞⁡Pσ+iTj=limj→∞⁡∑m=1ncmewmσ+iTj=limj→∞⁡∑m=1ncmewmσewmTjiexm-Tjwmi=limj→∞⁡∑m=1ncmewmσexmi=FPσ,x=0.Now, by Lemma 4, σ∈RP and then the theorem follows.All of this results in a second pointwise characterization, through geometrical principle (35), of the closure set of the real parts of the zeros of exponential polynomials of form (1).Theorem 6.
LetP(z)=c1ew1z+⋯+cnewnz be an exponential polynomial of type (1) with n>2. Then a real number σ0∈RP if and only if(35)cjeσ0wj≤∑k=1,k≠jnckeσ0wk,j=1,2,…,n.Proof.
Suppose thatσ0∈RP, then there exists some vector x∈Rn such that FP(σ0,x)=0 or, equivalently, ∑m=1n‍cmewmσ0exmi=0. Therefore, (36)cjewjσ0exji=-∑k=1,k≠jnckeσ0wkexki,j=1,2,…,nand, by taking the modulus, we get (37)cjewjσ0≤∑k=1,k≠jnckeσ0wk,j=1,2,…,n.
Conversely, suppose that the positive real numbers|ck|eσ0wk, k=1,2,…,n, satisfy inequalities (35). We index them in decreasing order so that m1 is such that |cm1|eσ0wm1≔max⁡{|ck|eσ0wk:k=1,2,…,n}, m2 such that |cm2|eσ0wm2≔max⁡{|ck|eσ0wk:k=1,2,…,n,k≠m1}, and so forth. Thus there is at least one n-sided polygon whose sides have these lengths [12, p. 71]. That means that there exist real numbers θ1,θ2,…,θn verifying (38)∑k=1nckeσ0wkeiθk=0.Consequently, by taking x∈Rn as the vector so that xk=θk+arg(ck) for k=1,…,n, where arg(ck) denotes the principal argument of ck, we have (39)FPσ0,x≔∑k=1nckewkσexki=0.Hence, from Theorem 5, σ0∈RP.Note that the proof of the result above includes a generalization of Euclid’s theorem on the inequality of the triangle, which leads to geometrical principle (35) that characterizes the sets RP.Remark 7.
Alternatively, as one of the referees pointed out, the necessity in Theorem6 follows from the following fact: if (35) is not true then there exists, by continuity, a number ε>0 such that the condition of type (35) is still false for the interval (σ0-ε,σ0+ε) and thus, by the triangle inequality, there are no zeros with real part in this interval, which means that σ0∉RP.
In any case, note that Theorem6 has been proved without using [8] (in particular [8, Theorem  9]).


## 4. The Description of the SetRP

As it will be also shown below, the number of gaps that can have the setRP defined by Rez:P(z)=0¯, associated with an exponential polynomial P(z)=c1ew1z+⋯+cnewnz of type (1) with n>2, depends on the number of real solutions of the n-2 intermediate equations:(40)cjewjσ=∑k=1,k≠jnckewkσ,j=2,…,n-1.More so, as the proof of the following lemma shows (compare with [8, Theorem3]), these equations above for the cases j=1 and j=n provide us with the extremes of the critical interval IP=[aP,bP], where aP and bP are defined in (2) and (3), respectively.Lemma 8.
LetP(z)=c1ew1z+⋯+cnewnz be an exponential polynomial of type (1) with n>2. The inequations c1eα1σ≤∑k=2nckeαkσ and cneαnσ≤∑k=1n-1ckeαkσ are satisfied for all σ in the critical interval [aP,bP]. Moreover, there exist σ1>aP and σ2<bP such that the intervals [aP,σ1] and [σ2,bP] are both contained in RP.Proof.
By defining the real functions(41)Aσ≔c1ew1σ-∑k=2nckewkσ,Bσ≔cnewnσ-∑k=1n-1ckewkσ,the number of changes of the sign of its coefficients, say WA and WB, is 1. Then if NA and NB are the numbers of zeros of A(σ) and B(σ), respectively, by Pólya and Szegő’s result [27, p. 46], WA-NA and WB-NB are even nonnegative integers. Hence, NA=NB=1 and we denote by x0P and x1P the unique simple zeros of A(σ) and B(σ), respectively. Then, since c1ew1x0P=∑k=2n‍ckewkx0P, we deduce that cjewjx0P<c1ew1x0P for all j=2,…,n and (42)cjewjx0P<∑k=1,k≠jnckewkx0P,j=2,…,n.Analogously, we can deduce that (43)cjewjx1P<∑k=1,k≠jnckewkx1P,j=1,…,n-1.Therefore, according to Theorem 6, x0P and x1P must coincide with aP and bP. Now, from continuity applied on the above strict inequalities we can determine two real numbers σ1 and σ2 such that any σ of the corresponding intervals [aP,σ1] and [σ2,bP] satisfies the n inequalities (35). Furthermore, A(σ) and B(σ) are nonpositive real numbers when σ is in the critical interval. The lemma is proved.We are now ready to give the description of the setRP≔Rez:P(z)=0¯ associated with an exponential polynomial P(z) of type (1). This leads us to another shorter proof of [8, Theorem9].Theorem 9.
Given an exponential polynomialP(z) of type (1) with n>2, RP is either aP,bP or the union of at most n-1 disjoint nondegenerate closed intervals. In the latter case, the gaps of RP are exclusively produced by those equations (40) having two solutions.Proof.
Note that, fixingσ∈IP, if we suppose that some of inequations (35) are not satisfied, then we deduce from Theorem 6 that σ∉RP. Thus, in view of Lemma 8, we next focus our attention on the n-2 intermediate inequations: (44)cjewjσ≤∑k=1,k≠jnckewkσ,j=2,…,n-1.Let us define the real functions:(45)Ajσ≔∑k=1j-1ckewkσ-cjewjσ+∑k=j+1nckewkσ,j=2,…,n-1.Since the number of changes of sign of their coefficients is 2, by Pólya and Szegő’s result [27, p. 46], each equation Aj(σ)=0 can have 0 solutions, 1 solution when the zero of Aj(σ) is of second order, or 2 (distinct) solutions, σj-<σj+, when Aj(σ) has two simple zeros. Furthermore, since Aj(σ) is clearly positive for |σ| sufficiently large, then Aj(σ)<0 is uniquely satisfied in the open intervals of the form (σj-,σj+) (i.e., when Aj(σ)=0 has two distinct solutions). Now, by defining the set (46)B≔j∈2,…,n-1:Ajσ=0has  two  solutionsσj-<σj+,we have, by continuity, σk+<σl- if k,l∈B with k<l. Define also the set (47)GRP≔⋃j∈Bσj-,σj+;then the complementary set of RP is (48)RPC=-∞,aP∪GRP∪bP,∞;that is, it consists of at most n disjoint open intervals and consequently RP consists of at most n-1 nondegenerate closed intervals.As a consequence of the theorem above, the setRP has no isolated point for any exponential polynomial of type (1) with n>2. The following important result is another easy consequence of this characterization theorem.Corollary 10.
Given an exponential polynomialP(z)=c1ew1z+⋯+cnewnz of type (1), RP is invariant with respect to the moduli of its coefficients cj′s.Example 11.
Consider the exponential polynomials:(49)PAz=ezlog⁡2+10ezlog⁡11+75·ezlog⁡23+15ezlog⁡37+ezlog⁡53,PBz=ezlog⁡2-10ezlog⁡11+75ezlog⁡23-15ezlog⁡37+ezlog⁡53whose frequencies {log⁡2,log⁡11,log⁡23,log⁡37,log⁡53} are linearly independent over the rationals. From Corollary 10, first notice that RPA=RPB. In this respect, the distribution of their zeros with imaginary part between 0 and 1000 can be observed in Figure 1, which was performed using Maple.Figure 1
Zeros of the functionsPA(z) and PB(z) of Example 11.Furthermore, by using Theorem9, we can exactly determine the boundary points of the closure set of the real parts of their zeros. In fact, the gaps of RPA=RPB are produced when we find two solutions in some of the following equations:(50)10·11σ=2σ+75·23σ+15·37σ+53σ,(51)75·23σ=2σ+10·11σ+15·37σ+53σ,(52)15·37σ=2σ+10·11σ+75·23σ+53σ.Equation (50) has no solutions. Equality (51) is reached in σ1≈-1.48326 and σ2≈2.97998. Finally, σ3≈4.13163 and σ4≈6.97891 are the solutions of (52). Also, from Lemma 8, we calculate the extremes of the critical interval which are a≈-1.9734 and b≈7.8502. Therefore, the disjoint nondegenerate closed intervals of RPA=RPB are [a,σ1], [σ2,σ3], and [σ4,b].


## 5. On the Converse of the Result of Invariance

Given an exponential polynomialP(z)=c1ew1z+c2ew2z+⋯+cnewnz of form (1), by dividing by the first term, it is not restrictive to consider, for our purposes, its underlying exponential polynomial: (53)PNz=1+m1eα1z+⋯+mn-1eαn-1z,where mj=cj+1/c1 and αj=wj+1-w1 for j=1,2,…,n-1.We are now going to study whether the converse of Corollary10 is true; that is, given two exponential polynomials P(z) and Q(z) of type (1) with the same set of frequencies and so that RP=RQ, are the moduli of the coefficients mj (in the normalized form PN(z) and QN(z)) the same? It is an elementary check that the answer is affirmative when P(z) and Q(z) have two terms. We next prove that the answer is also affirmative when our exponential polynomials have three terms (the case n=3).Proposition 12.
LetP(z)=c1ew1z+c2ew2z+c3ew3z and Q(z)=d1ew1z+d2ew2z+d3ew3z be two exponential polynomials of type (1) and let PN(z)=1+m1eα1z+m2eα2z and QN(z)=1+s1eα1z+s2eα2z be their normalized forms. Thus RP=RQ if and only if |mj|=|sj| for each j=1,2.Proof.
If|mj|=|sj| for j=1,2 then, by taking Corollary 10 into account, we easily get RP=RQ. Conversely suppose that RP=RQ; then in particular we have that aP=aQ and bP=bQ. Hence, we deduce from the proof of Lemma 8 that (54)c1ew1aP=c2ew2aP+c3ew3aP,c3ew3bP=c1ew1bP+c2ew2bP,d1ew1aP=d2ew2aP+d3ew3aP,d3ew3aP=d1ew1bP+d2ew2bP.Equivalently, (55)1=m1eα1aP+m2eα2aP,m2eα2bP=1+m1eα1bP,1=s1eα1aP+s2eα2aP,s2eα2bP=1+s1eα1bP.Therefore, it is an elementary check that (56)m1=-eα2aP+eα2bPeα1aPeα2bP+eα2aPeα1bP=s1,m2=eα1aP+eα1bPeα1aPeα2bP+eα2aPeα1bP=s2,and the result follows.However, as the following example shows, Proposition12 cannot be extended to any integer number n>3.Example 13.
LetPN(z)=1+m12z+m23z+m35z and QN(z)=1+s12z+s23z+s35z be the normalized exponential polynomials associated with two exponential polynomials P(z) and Q(z) of type (1) and with the same frequencies, where m1=m2=m3=1, and (57)s1≔5bP-2·3aP5bP-5aP-2·3bP5aP2aP5bP+2bP5aP≈0.2316055032,s2=2, and (58)s3≔2aP+2bP+2·2aP3bP-2·2bP3aP2aP5bP+2bP5aP≈1.284781118.Notice that |mj|≠|sj| for j=1,2,3. Nevertheless, it is verified that RP=RQ. Indeed, s1, s2, and s3 have been chosen so that the extremes of the critical interval verify aP=aQ and bP=bQ or, equivalently, they satisfy (59)1=s12aP+s23aP+s35aP,s35bP=1+s12bP+s23bP.More so, (40) have no solution for both exponential polynomials and thus RP=RQ=[aP,bP], where aP≈-1.03281 and bP≈1.22446. That is, RP=RQ has no gaps. In this respect, the distribution of the zeros of PN(z) and QN(z) (which are the same as P(z) and Q(z), resp.) whose imaginary part is between -5000 and 5000 can be observed in Figure 2.Figure 2
Zeros of the functionsP(z) and Q(z) of Example 13.



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