A discrete set in the $p$-dimensional Euclidian space is {\it almost
periodic}, if the measure with the unite masses at points of the set is almost
periodic in the weak sense. We propose to construct positive almost periodic
discrete sets as an almost periodic perturbation of a full rank discrete
lattice. Also we prove that each almost periodic discrete set on the real axes
is an almost periodic perturbation of some arithmetic progression.
  Next, we consider signed almost periodic discrete sets, i.e., when the signed
measure with masses $\pm1$ at points of a discrete set is almost periodic. We
construct a signed discrete set that is not almost periodic, while the
corresponding signed measure is almost periodic in the sense of distributions.
Also, we construct a signed almost periodic discrete set such that the measure
with masses +1 at all points of the set is not almost periodic.Comment: 6 page

Favorov, S.

Kolbasina, Ye.

Algebra and Discrete Mathematics

English

arXiv

A discrete set in the p-dimensional Euclidian space is  almost periodic, if  the measure with the unite masses at points of the set is almost periodic in the weak sense. We propose to construct positive almost periodic discrete sets as an almost periodic perturbation of a full rank discrete lattice. Also we prove that each almost periodic discrete set on the real axes is an almost periodic perturbation of some arithmetic progression.



Next,  we consider signed almost  periodic discrete sets, i.e., when the signed measure with masses +1 or -1 at points of a discrete set is almost periodic. We construct a signed discrete set that is not almost periodic, while the corresponding signed measure is almost periodic in the sense of distributions. Also, we construct a signed almost periodic discrete set such that the measure with masses +1 at all points of the set is not almost periodic

Kolbasina, Y.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

ADMDRAFTAlgebra and Discrete Mathematics RESEARCH ARTICLEVolume 9 (2010). Number 2. pp. 48 – 58c© Journal “Algebra and Discrete Mathematics”Perturbations of discrete latticesand almost periodic setsFavorov Sergey and Kolbasina YevgeniiaCommunicated by B. V. NovikovAbstract. A discrete set in the p-dimensional Euclidianspace is almost periodic, if the measure with the unite masses atpoints of the set is almost periodic in the weak sense. We propose toconstruct positive almost periodic discrete sets as an almost periodicperturbation of a full rank discrete lattice. Also we prove that eachalmost periodic discrete set on the real axes is an almost periodicperturbation of some arithmetic progression.Next, we consider signed almost periodic discrete sets, i.e., whenthe signed measure with masses +1 or -1 at points of a discrete setis almost periodic. We construct a signed discrete set that is notalmost periodic, while the corresponding signed measure is almostperiodic in the sense of distributions. Also, we construct a signedalmost periodic discrete set such that the measure with masses +1at all points of the set is not almost periodic.The concept of almost periodicity plays an important role in variousbranches of analysis. In particular, almost periodic discrete sets are usedfor investigation of zero sets of some holomorphic functions (cf. [8],[5]),in value distribution theory of some classes of meromorphic functions(cf. [3]), as a model of quasicrystals (cf. [7],[9]). Note that in [7] thequestion (Problem 4.4) was raised if there exist other discrete almostperiodic sets in Rp, besides of the form L+ E with a discrete lattice Land a finite set E.Here we propose several ways to construct almost periodic discretesets. Next, we introduce signed almost periodic discrete sets. In particular,2000 Mathematics Subject Classification: 11K70; 52C07, 52C23.Key words and phrases: perturbation of discrete lattice, almost periodic discreteset, signed discrete set, quasicrystals.ADMDRAFTS. Favorov, Ye. Kolbasina 49we construct signed discrete set such that its associate measure is almostperiodic in the sense of distributions and not almost periodic in the weaksense.To formulate our result beforehand we have to recall some knowndefinitions (see, for example, [1]).A continuous function f(x) in Rp is almost periodic, if for any ε > 0the set of ε-almost periods of f{τ ∈ Rp : supx∈Rp|f(x+ τ)− f(x)| < ε}is a relatively dense set in Rp. The latter means that there is R = R(ε) <∞such that any ball of radius R contains an ε-almost period of f .Note that almost periodic functions are uniformly bounded in Rp.Besides, every almost periodic function is the uniform in x ∈ Rp limit ofa sequence of exponential polynomials of the formP (x) =∑mcmei〈x,λm〉, λm ∈ Rp, cm ∈ C, (1)here 〈., .〉 is the scalar product in Rp.A Borel measure µ in Rp is almost periodic if it is almost periodic inthe weak sense, i.e., for any continuous function ϕ in Rp with a compactsupport the convolution ∫ϕ(x+ t) dµ(t) (2)is an almost periodic function in x ∈ Rp (see [10]).The definition suits to signed measures as well.Theorem 1 ([10], Theorems 2.1 and 2.7). For any signed almost periodicmeasure µ in Rp there exists M <∞ such that the variation |µ| satisfiesthe condition|µ|(B(c, 1)) < M ∀c ∈ Rp. (3)Besides, there exists uniformly in x ∈ Rp a finite limitD(µ) = limR→∞µ(B(x,R))ωpRp.Here B(x,R) is an open ball with the center at the point x and radiusR, ωp is the volume of B(0, 1).Following [7], we will say that a discrete set A is a Delone set, if thereare r > 0 and R <∞ such that each ball of radius r contains at most oneelement of A, and each ball of radius R contains at least one element of A.ADMDRAFT50 Perturbations of discrete latticesDefinition 1 ([7]). A Delone set A is almost periodic, if its associatemeasureµA =∑x∈Aδx, (4)where δx is the unit mass at the point x, is almost periodic.We will consider some generalization of discrete sets, namely multiplediscrete sets in Rp. This means that a number m(x) ∈ N corresponds toeach point x from a discrete set. We denote this object by A = {(x,m(x))}and the corresponding discrete set by s(A). Also, we will write A = (ak),where every term a ∈ s(A) appears m(a) times in the sequence (ak).In the case p = 2 the definition coincides with the definition of thedivisor of an entire function in the complex plane.The definition of almost periodic Delone sets has an evident general-ization to almost periodic multiple discrete set s.Definition 2. A multiple discrete set A is almost periodic, if its associatemeasureµA =∑x∈s(A)m(x)δx (5)is almost periodic.Putcard(A ∩ E) =∑x∈s(A)∩Em(x)for any E ⊂ Rp. The following result is a consequence of Theorem 1.Theorem 2. For any almost periodic multiple discrete setA there existsM <∞ such thatcard (A ∩B(x, 1)) < M ∀x ∈ Rp. (6)Besides, there exists uniformly in x ∈ Rp a finite densityD(A) = limR→∞card(A ∩B(x,R))ωpRp. (7)Another proof of Theorem 2 see in [4].There is a geometric criterium for multiple discrete sets to be almostperiodic.ADMDRAFTS. Favorov, Ye. Kolbasina 51Theorem 3 ([4], Theorem 11). An almost periodic multiple discrete set(an) ⊂ Rp is almost periodic if and only if for each ε > 0 the set of ε-almostperiods of (an){τ ∈ Rp : ∃ a bijectionσ : N→ N such that supn∈N|an + τ − aσ(n)| < ε}(8)is relatively dense in Rp.At the first time almost periodic divisors appeared in papers [8] and[11], where only shifts along real axis were considered. The definition ofalmost periodicity based on the above geometric property. An analog ofTheorem 3 was proved in [5].Almost periodic perturbations of discrete lattices. Let F (x) =(F1(x), . . . , Fp(x)) be a mapping from Rp to Rp with almost periodiccomponents Fj(x). For convenience of a reader, prove the following knownassertion.Proposition. For any ε > 0 the set of common ε-almost periods of Fjwith integer coordinates is relatively dense in Rp.Proof. By the well-known Kronecker Theorem, the system of inequalities| exp〈τ, λn〉 − 1| < δ, n = 1, . . . , N, (9)has a relatively dense in Rp set of solutions τ for any δ > 0 and anyλn ∈ Rp, n = 1, . . . , N . Let {ej}pj=1 be the natural basis in Rp. Commonsolutions of (9) and the system| exp 2pi〈τ, ej〉 − 1| < η, j = 1, . . . , p, (10)form a relatively dense set too. Whenever τ satisfies (10), there is r ∈ Zpsuch that |τ − r| < pη/pi. Hence for sufficiently small η there exists arelatively dense set of solutions r ∈ Zp of system (9) with 2δ instead of δ.If each function Fj is an exponential polynomial of form (1), then thereexist δ and λ1, . . . , λN such that these solutions are common ε-almostperiods of Fj . In the general case we can approximate the functions Fj(x)by sequences of exponential polynomials.Let F be the same as above, and L be an arbitrary discrete full ranklattice in Rp. Rewrite it in the form L = {kΓ, k ∈ Zp}, where Γ is anon-generated p × p matrix. If r ∈ Zp is a common ε-almost period ofADMDRAFT52 Perturbations of discrete latticescomponents of F , then τ = rΓ is an pε-almost period of the discretemultiple setA = (ak), ak = kΓ + F (k), k ∈ Zp, (11)where the bijection σ : Zp → Zp in (8) has the form σ(k) = k + r.Whenever all components F are almost periodic, Theorem 3 implies thatA is an almost periodic multiple discrete set. Also, note that in the caseof sufficiently small supRp |F (x)| we obtain an almost periodic Delone set.It is easy to construct an almost periodic set in Rp without anyperiods. Take F (x) = 13√p(sinx1, . . . , sinxp) for x = (x1, . . . , xp), andput A = k + F (k), k ∈ Zp. If τ ∈ Rp \ {0} is a period of A, thenk + τ + F (k) = k′ + F (k′) for all k ∈ Zp and some k′ = k′(k, τ). Clearly,τ = τ (1) + τ (2), where τ (1) = (τ(1)1 , . . . , τ(1)p ) ∈ Zp and modula of allcomponents of τ (2) = (τ(2)1 , . . . , τ(2)p ) is less then 1/2. Therefore, k′ =k+ τ (1) and sin(kj + τ(1)j ) = τ(2)j + sin kj for all k = (k1, . . . , kp) ∈ Zp andj = 1, . . . , p, that is impossible.Clearly, all vectors from a lattice L are periods of every almost periodicset of the form L+ E with a finite set E. Therefore we obtain an answeron the question raised in [7].Note that in [2] and [6] we have got representation (11) with a squirelattice L and a bounded mapping F (x) for a wide class of multiple discretesets in Rp, in particular, for every almost periodic multiple discrete set s.We do not know if each almost periodic set has representation (11) withsome lattice L and a mapping F (x) with almost periodic coordinates Fj .But this is true for almost periodic sets in the real axis.Theorem 4. Let A = (ak)k∈Z, where ak ≤ ak+1 for all k, be an almostperiodic multiple discrete set in R with the density D. Then ak = Dk+f(k)with an almost periodic function f .In [8] a similar result was obtained for real parts of zeros of almostperiodic entire functions from some special class.Proof. Without loss of generality we may suppose that density D of theset A is equal to 1. Also we suppose that 0 ∈ A and a0 = 0. Take arbitraryx, y, h ∈ R, x < y, h > 0. Using Theorem 3, take L > 2 such thatany interval i ⊂ R of length L contains a 1-almost period κ of A. Sincei− κ ⊂ (−L, L), we getcard(A ∩ i) ≤M, where M = card(A ∩ (−L− 1, L+ 1)).ADMDRAFTS. Favorov, Ye. Kolbasina 53Take a 1-almost period κ of the set A such that y < x+ κ < y + L. Bydefinition, there is a bijection ρ between all points of the set A∩(x+κ, x+κ+ h] and some points of the set A ∩ (x− 1, x+ h+ 1]. Moreover, thesame ρ is a bijection between some points of the set A∩ (x+κ, x+κ+h]and all points of the set A ∩ (x+ 1, x+ h− 1]. Therefore we have|card(A ∩ (x+ κ, x+ κ+ h])− card(A ∩ (x, x+ h])|≤ card(A ∩ ((x− 1, x+ 1] ∪ (x+ h− 1, x+ h+ 1]) ≤ 2M.Since(x+ κ, x+ κ+ h] \ (y, y + h] ⊂ (y + h, y + h+ L),(y, y + h] \ (x+ κ, x+ κ+ h] ⊂ (y, y + L),we obtain|card(A ∩ (x, x+ h])− card(A ∩ (y, y + h])|≤ 2M + card(A ∩ [(y, y + L) ∪ (y + h, y + h+ L)]) ≤ 4M.For any T ∈ N the half–interval (x, x+ Th] is the union of half–intervals(x+(j−1)h, x+jh], j = 1, . . . , T . If we set y = x+(j−1)h, j = 2, . . . , T ,we get|card(A ∩ (x, x+ h])− T−1card(A ∩ (x, x+ Th])| ≤ 4M.Taking into account (7) with D(A) = 1, we obtain|card(A ∩ (x, x+ h])− h| ≤ 4M ∀x ∈ R, h > 0. (12)Furthermore, putn(t) = card(A ∩ (0, t]) for t > 0,n(t) = −card(A ∩ (t, 0]) for t < 0, n(0) = 0.Take ε < L/(24M). Let τ > 2L be an ε-almost period of the set A. Clearly,if x, y, x + τ, y + τ do not belong to the ε-neighborhood Uε of the sets(A), then we haven(y + τ)− n(x+ τ) = n(y)− n(x).Therefore, the function n(x+ τ)− n(x) takes the same number p ∈ N forall x ∈ R \ (Uε ∪ (Uε − τ)).Next, denote by E[a] the integer part of a real number a. Put N =E[L/(4Mε)] + 1. It is easily shown thatL4Mε< N <τ2Mε(E[τ/L] + 1)− 1. (13)ADMDRAFT54 Perturbations of discrete latticesDenote bymesG the Lebesgue measure of the set G. Since any half-interval(y, y + τ ] contains at most M(E[τ/L] + 1) points of the set A, we getmes N⋃j=0[(Uε − jτ) ∩ (0, τ)] ≤N∑j=0mes[Uε ∩ (jτ, (j + 1)τ)]≤ (N + 1)2εM(E[τ/L] + 1) < τ.Hence there is x ∈ (0, τ) such that the points x, x+ τ, . . . , x+Nτ donot belong to Aε. Therefore,n(x+Nτ)− n(x) =N∑j=1n(x+ jτ)− n(x+ (j − 1)τ) = Np.On the other hand, using (12) with h = Nτ , we get|n(x+Nτ)− n(x)−Nτ | < 4M.Consequently, by (13), we get |τ − p| < 4M/N < 16M2ε/L.Put γ(k) = ak − k for all k ∈ Z. We shall prove that|γ(m+ p)− γ(m)| < Hε ∀m ∈ Z, (14)with H = 5M + 16M2/L. Suppose the contrary. For example, let γ(m+p) > γ(m) +Hε for some m ∈ Z. This yields thatam+p > am + p+Hε > am + τ + 5Mε.Since an ≤ am for n < m and an ≥ am+p for n > m+ p, we see that forall t ∈ (am, am + 5Mε) we haven(t+ τ)− n(t) = card(A ∩ (t, t+ τ ]) ≤ p− 1. (15)On the other hand,mes((am, am + L) ∩ [Uε ∪ (Uε − τ)])≤ 2ε card([am, am + L] ∩ [A ∪ (A− τ)]) ≤ 4Mε < 5Mε.Since the left-hand side of (15) is equal to p for all t ∈ (am, am + L) \[Uε ∪ (Uε − τ)], we obtain a contradiction. In the same way we prove thatthe case γ(m+ p) < γ(m)−Hε is impossible as well. Hence (14) is validfor all m ∈ Z. If we continue the function γ as a linear function to eachinterval (m, m+ 1), we obtain the continuous function f on R such that|f(x+ p)− f(x)| < Hε, ∀x ∈ R.Since the number p with this property exists in the (16M2ε/L)-neighbor-hood of each ε-almost period τ of the set A, we see that f is an almostperiodic function. Theorem is proved.ADMDRAFTS. Favorov, Ye. Kolbasina 55Signed multiple discrete sets. Now we will consider some general-ization of discrete sets, namely signed multiple discrete sets in Rp. Thismeans that a number m(x) ∈ Z \ {0} corresponds to each point x froma discrete set. As above, we denote this object by A = {(x,m(x))} andthe corresponding discrete set by s(A). Equality (5) define the associatemeasure µA of A. Also, putA+ = {(x,m(x)), x ∈ s(A), m(x) > 0},A− = {(x,m(x)), x ∈ s(A), m(x) < 0}.In the case p = 2 the definition coincides with the definition of thedivisor of a meromorphic function in the complex plane.Definition 3. A signed multiple discrete set A is almost periodic, if itsassociate measure µA is almost periodic.Note that each continuous function with a compact support can beapproximated by a sequence of functions from C∞ with supports in afixed ball. Therefore, if a signed measure µ satisfies (3), we can take onlyfunctions ϕ ∈ C∞ in definition (2). Next, take a positive function ϕ ∈ C∞such that ϕ(x) ≡ 1 for |x| ≤ 1 and ϕ(x) ≡ 0 for |x| ≥ 2 in (2). Sincealmost periodic functions are bounded in Rp, we see that every almostperiodic in the sense of distributions positive measure satisfies (3). Hencethe class of positive multiple discrete sets with almost periodic in the senseof distributions associate measures coincides with the class of positivealmost periodic multiple discrete sets. But this assertion does not validfor signed multiple discrete sets.Theorem 5. There is a signed multiple discrete set such that its associatemeasure is almost periodic in the sense of distributions and not almostperiodic in the weak sense.Proof. Let α(n), n ∈ 2Z \ {0}, be the greatest k ∈ N such that 2k is adivisor of n. Puta+n = n+ 1/(α(n) + 1)2, a−n = n− 1/(α(n) + 1)2, n ∈ 2Z \ {0}.Define the signed multiple discrete set A = A+ ∪A−, whereA+ = {(a+n , α(n))}n∈2Z\{0}, A− = {(a−n , α(n))}n∈2Z\{0}.The measure µA does not satisfy (3), therefore it is not almost periodic.Let us show that µA is almost periodic in the sense of distributions.ADMDRAFT56 Perturbations of discrete latticesTake a function ϕ ∈ C∞ such that suppϕ ⊂ (−1/2, 1/2). Supposethat τ = 2pk for some p ∈ N, k ∈ Z. If |x− n| ≥ 3/4 for all n ∈ 2Z, thenthe same is valid for the point x+ τ , therefore,(ϕ ∗ µA)(x+ τ) = (ϕ ∗ µA)(x) = 0.If |x−n| < 3/4 for n ∈ {2Z : α(n) ≥ p}∪{0}, then either (ϕ∗µA)(x) =0, or|ϕ ∗ µA(x)| = α(n)|ϕ(a+n + x)− ϕ(a−n + x)| ≤ α(n)M |a+n − a−n | < 2M/p,where M = supR |ϕ′(x)|. Moreover, if this is the case, then also n+ τ ∈{2Z : α(n) ≥ p} ∪ {0}. Hence the same bound is valid for the value(ϕ ∗ µA)(x+ τ). We obtain|ϕ ∗ µA(x)− ϕ ∗ µA(x+ τ)| < 4M/p.Finally, if |x−n| < 3/4 for n ∈ {2Z : α(n) < p}, then α(n+τ) = α(n).Therefore, we have a±n+τ = a±n + τ , and(ϕ ∗ µA)(x+ τ) = α(n+ τ)[ϕ(x+ τ − a+n+τ )− ϕ(x+ τ − a−n+τ )]= α(n)[ϕ(x− a+n )− ϕ(x− a−n )] = (ϕ ∗ µA)(x).Consequently, all multiplies of 2p are (4M/p)-almost periods of the function(ϕ ∗ µA)(x). Thus µA is almost periodic in the sense of distributions.Theorem is proved.Note that we can check almost periodicity of function (2) only forpositive continuous functions ϕ with an arbitrary small diameter of itssupport. Hence if A is a signed almost periodic multiple discrete set andinf{|x− y| : x ∈ s(A+), y ∈ s(A−)} > 0,then A+ and A− are almost periodic multiple discrete set s as well. Butthis is false in the general case.Theorem 6. There is a signed almost periodic set A such that A+ andA− are not almost periodic.Proof. PutA+ = {(a+n , 1)}n∈2Z, A− = {(a−n , −1)}n∈2Z, A = A+ ∪A−,where points a±n are the same as in the proof of Theorem 5.ADMDRAFTS. Favorov, Ye. Kolbasina 57Following the proof of Theorem 5, we can assure that µA is almostperiodic in the sense of distributions. Since the measure µA satisfiescondition (3), we get that A is a signed almost periodic multiple discreteset.We will use Theorem 3 for proving that A+ is not almost periodic.Clearly, the distance between any two points of A+ has the form 2m+β, m ∈ N, |β| < 1/4. Hence whenever τ is an ε-almost period of A+,ε < 1/4, we have τ = 2n0 + γ, n0 ∈ Z, |γ| < 1/2. But 0 6∈ A+, hence thedistance between the point 2n0 + (α(2n0) + 1)−2 − τ and any point ofA+ is more than 1. We obtain a contradiction. Consequently, A+ is notalmost periodic. Analogously, A− is not almost periodic as well. Theoremis proved.Clearly, the measure |µA| = 2µA+ − µA is not almost periodic as well.Therefore we obtainCorollary. The positive discrete set {a+n }∪{a−n } does not almost periodic.References[1] C.Corduneanu, Almost Periodic Functions, Interscience Publishers, New-York –London – Sydney – Toronto, a division of John Wiley.[2] A. Dudko, S. Favorov, A uniformly spread measure criterion, Preprint,arXiv:0805.0999.[3] S.Favorov, Sunyer-i-Balaguer’s Almost Elliptic Functions and Yosida’s NormalFunctions, J. d’Analyse Math.,Vol.104 (2008), 307-340.[4] S. Favorov, Ye. Kolbasina Almost periodic discrete sets, Journal of MathematicalPhysics, Analysis, Geometry. (2010), vol.6, No.1.[5] S.Yu. Favorov, A.Yu.Rashkovskii, A.I. Ronkin, Almost periodic divisors in a strip,J. d’Analyse Math., Vol 74 (1998), 325-345.[6] Ye. Kolbasina, On the property of discrete sets in Rk, Visn. Khark. Univ., Ser. Mat.Prykl. Mat. Mekh., 2008, N.826, p.52-66 (Russian).[7] J.C. Lagarias, Mathematical quasicristals and the problem of diffraction, Directionsin Mathematical Quasicrustals, M. Baake and R. Moody, eds., CRM Monographseries, Vol. 13, AMS, Providence RI, 2000, 61-93.[8] B.Ja. Levin,Distributions of Zeros of Entire Functions. Transl. of Math. Monograph,Vol.5, AMS Providence, R1, 1980.[9] R.V. Moody,M. Nesterenco, and J. Patera,Computing with almost periodic functions,Preprint, arXiv:0808.1814v1 [math-ph] 13 Aug 2008.[10] L.I. Ronkin, Almost periodic distributions and divisors in tube domains, Zap.Nauchn. Sem. POMI 247 (1997), 210-236 (Russian).[11] H.Tornehave, Systems of zeros of holomorphic almost periodic functions, Koben-havns Universitet Matematisk Institut, Preprint No. 30, 1988, 52 p.ADMDRAFT58 Perturbations of discrete latticesContact informationS. Favorov Mathematical School, Kharkov NationalUniversity, Swobody sq.4, Kharkov, 61077UkraineE-Mail: Sergey.Ju.Favorov@univer.kharkov.uaURL: www-mechmath.univer.kharkov.ua/funcan/staff/favorov/index.htmlYe. Kolbasina Mathematical School, Kharkov NationalUniversity, Swobody sq.4, Kharkov, 61077UkraineE-Mail: kvr_jenya@mail.ruURL: -Received by the editors: 12.02.2010and in final form ????.

Perturbations of discrete lattices and almost periodic sets

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