Perturbations of discrete lattices and almost periodic sets
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....
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| Опубліковано в: : | Algebra and Discrete Mathematics |
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| Дата: | 2010 |
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Інститут прикладної математики і механіки НАН України
2010
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Perturbations of discrete lattices and almost periodic sets / S. Favorov, Y. Kolbasina // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 2. — С. 48–58. — Бібліогр.: 11 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860239670939484160 |
|---|---|
| author | Favorov, S. Kolbasina, Y. |
| author_facet | Favorov, S. Kolbasina, Y. |
| citation_txt | Perturbations of discrete lattices and almost periodic sets / S. Favorov, Y. Kolbasina // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 2. — С. 48–58. — Бібліогр.: 11 назв. — англ. |
| collection | DSpace DC |
| container_title | Algebra and Discrete Mathematics |
| description | 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.
|
| first_indexed | 2025-12-07T18:28:35Z |
| format | Article |
| fulltext |
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Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 9 (2010). Number 2. pp. 48 – 58
c© Journal “Algebra and Discrete Mathematics”
Perturbations of discrete lattices
and almost periodic sets
Favorov Sergey and Kolbasina Yevgeniia
Communicated by B. V. Novikov
Abstract. 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.
The concept of almost periodicity plays an important role in various
branches of analysis. In particular, almost periodic discrete sets are used
for 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] the
question (Problem 4.4) was raised if there exist other discrete almost
periodic sets in R
p, besides of the form L+ E with a discrete lattice L
and a finite set E.
Here we propose several ways to construct almost periodic discrete
sets. 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 discrete
set, signed discrete set, quasicrystals.
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S. Favorov, Ye. Kolbasina 49
we construct signed discrete set such that its associate measure is almost
periodic in the sense of distributions and not almost periodic in the weak
sense.
To formulate our result beforehand we have to recall some known
definitions (see, for example, [1]).
A continuous function f(x) in R
p is almost periodic, if for any ε > 0
the set of ε-almost periods of f
{τ ∈ R
p : sup
x∈Rp
|f(x+ τ)− f(x)| < ε}
is a relatively dense set in R
p. 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 R
p.
Besides, every almost periodic function is the uniform in x ∈ R
p limit of
a sequence of exponential polynomials of the form
P (x) =
∑
m
cmei〈x,λm〉, λm ∈ R
p, cm ∈ C, (1)
here 〈., .〉 is the scalar product in R
p.
A Borel measure µ in R
p is almost periodic if it is almost periodic in
the weak sense, i.e., for any continuous function ϕ in R
p with a compact
support the convolution
∫
ϕ(x+ t) dµ(t) (2)
is an almost periodic function in x ∈ R
p (see [10]).
The definition suits to signed measures as well.
Theorem 1 ([10], Theorems 2.1 and 2.7). For any signed almost periodic
measure µ in R
p there exists M < ∞ such that the variation |µ| satisfies
the condition
|µ|(B(c, 1)) < M ∀c ∈ R
p. (3)
Besides, there exists uniformly in x ∈ R
p a finite limit
D(µ) = lim
R→∞
µ(B(x,R))
ωpRp
.
Here B(x,R) is an open ball with the center at the point x and radius
R, ωp is the volume of B(0, 1).
Following [7], we will say that a discrete set A is a Delone set, if there
are r > 0 and R < ∞ such that each ball of radius r contains at most one
element of A, and each ball of radius R contains at least one element of A.
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50 Perturbations of discrete lattices
Definition 1 ([7]). A Delone set A is almost periodic, if its associate
measure
µA =
∑
x∈A
δx, (4)
where δx is the unit mass at the point x, is almost periodic.
We will consider some generalization of discrete sets, namely multiple
discrete sets in R
p. This means that a number m(x) ∈ N corresponds to
each 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 the
divisor 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 associate
measure
µA =
∑
x∈s(A)
m(x)δx (5)
is almost periodic.
Put
card(A ∩ E) =
∑
x∈s(A)∩E
m(x)
for any E ⊂ R
p. The following result is a consequence of Theorem 1.
Theorem 2. For any almost periodic multiple discrete setA there exists
M < ∞ such that
card (A ∩B(x, 1)) < M ∀x ∈ R
p. (6)
Besides, there exists uniformly in x ∈ R
p a finite density
D(A) = lim
R→∞
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 almost
periodic.
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S. Favorov, Ye. Kolbasina 51
Theorem 3 ([4], Theorem 11). An almost periodic multiple discrete set
(an) ⊂ R
p is almost periodic if and only if for each ε > 0 the set of ε-almost
periods of (an)
{τ ∈ R
p : ∃ a bijectionσ : N → N such that sup
n∈N
|an + τ − aσ(n)| < ε}
(8)
is relatively dense in R
p.
At the first time almost periodic divisors appeared in papers [8] and
[11], where only shifts along real axis were considered. The definition of
almost periodicity based on the above geometric property. An analog of
Theorem 3 was proved in [5].
Almost periodic perturbations of discrete lattices. Let F (x) =
(F1(x), . . . , Fp(x)) be a mapping from R
p to R
p with almost periodic
components Fj(x). For convenience of a reader, prove the following known
assertion.
Proposition. For any ε > 0 the set of common ε-almost periods of Fj
with integer coordinates is relatively dense in R
p.
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 R
p set of solutions τ for any δ > 0 and any
λn ∈ R
p, n = 1, . . . , N . Let {ej}
p
j=1 be the natural basis in Rp. Common
solutions of (9) and the system
| exp 2π〈τ, ej〉 − 1| < η, j = 1, . . . , p, (10)
form a relatively dense set too. Whenever τ satisfies (10), there is r ∈ Z
p
such that |τ − r| < pη/π. Hence for sufficiently small η there exists a
relatively dense set of solutions r ∈ Z
p of system (9) with 2δ instead of δ.
If each function Fj is an exponential polynomial of form (1), then there
exist δ and λ1, . . . , λN such that these solutions are common ε-almost
periods 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 rank
lattice in R
p. Rewrite it in the form L = {kΓ, k ∈ Z
p}, where Γ is a
non-generated p × p matrix. If r ∈ Z
p is a common ε-almost period of
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52 Perturbations of discrete lattices
components of F , then τ = rΓ is an pε-almost period of the discrete
multiple set
A = (ak), ak = kΓ + F (k), k ∈ Z
p, (11)
where the bijection σ : Z
p → Z
p in (8) has the form σ(k) = k + r.
Whenever all components F are almost periodic, Theorem 3 implies that
A is an almost periodic multiple discrete set. Also, note that in the case
of sufficiently small supRp |F (x)| we obtain an almost periodic Delone set.
It is easy to construct an almost periodic set in R
p without any
periods. Take F (x) = 1
3
√
p
(sinx1, . . . , sinxp) for x = (x1, . . . , xp), and
put A = k + F (k), k ∈ Z
p. If τ ∈ R
p \ {0} is a period of A, then
k + τ + F (k) = k′ + F (k′) for all k ∈ Z
p and some k′ = k′(k, τ). Clearly,
τ = τ (1) + τ (2), where τ (1) = (τ
(1)
1 , . . . , τ
(1)
p ) ∈ Z
p and modula of all
components 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) ∈ Z
p and
j = 1, . . . , p, that is impossible.
Clearly, all vectors from a lattice L are periods of every almost periodic
set of the form L+ E with a finite set E. Therefore we obtain an answer
on the question raised in [7].
Note that in [2] and [6] we have got representation (11) with a squire
lattice L and a bounded mapping F (x) for a wide class of multiple discrete
sets in R
p, in particular, for every almost periodic multiple discrete set s.
We do not know if each almost periodic set has representation (11) with
some 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 almost
periodic 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 almost
periodic entire functions from some special class.
Proof. Without loss of generality we may suppose that density D of the
set A is equal to 1. Also we suppose that 0 ∈ A and a0 = 0. Take arbitrary
x, y, h ∈ R, x < y, h > 0. Using Theorem 3, take L > 2 such that
any interval i ⊂ R of length L contains a 1-almost period κ of A. Since
i− κ ⊂ (−L, L), we get
card(A ∩ i) ≤ M, where M = card(A ∩ (−L− 1, L+ 1)).
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S. Favorov, Ye. Kolbasina 53
Take a 1-almost period κ of the set A such that y < x+ κ < y + L. By
definition, 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, the
same 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, put
n(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 set
s(A), then we have
n(y + τ)− n(x+ τ) = n(y)− n(x).
Therefore, the function n(x+ τ)− n(x) takes the same number p ∈ N for
all 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 that
L
4Mε
< N <
τ
2Mε(E[τ/L] + 1)
− 1. (13)
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54 Perturbations of discrete lattices
Denote by mesG 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 get
mes
N
⋃
j=0
[(Uε − jτ) ∩ (0, τ)]
≤
N
∑
j=0
mes[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 do
not belong to Aε. Therefore,
n(x+Nτ)− n(x) =
N
∑
j=1
n(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 that
am+p > am + p+Hε > am + τ + 5Mε.
Since an ≤ am for n < m and an ≥ am+p for n > m+ p, we see that for
all t ∈ (am, am + 5Mε) we have
n(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 that
the case γ(m+ p) < γ(m)−Hε is impossible as well. Hence (14) is valid
for all m ∈ Z. If we continue the function γ as a linear function to each
interval (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 almost
periodic function. Theorem is proved.
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S. Favorov, Ye. Kolbasina 55
Signed multiple discrete sets. Now we will consider some general-
ization of discrete sets, namely signed multiple discrete sets in R
p. This
means that a number m(x) ∈ Z \ {0} corresponds to each point x from
a discrete set. As above, we denote this object by A = {(x,m(x))} and
the corresponding discrete set by s(A). Equality (5) define the associate
measure µA of A. Also, put
A+ = {(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 the
divisor of a meromorphic function in the complex plane.
Definition 3. A signed multiple discrete set A is almost periodic, if its
associate measure µA is almost periodic.
Note that each continuous function with a compact support can be
approximated by a sequence of functions from C∞ with supports in a
fixed ball. Therefore, if a signed measure µ satisfies (3), we can take only
functions 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). Since
almost periodic functions are bounded in R
p, we see that every almost
periodic in the sense of distributions positive measure satisfies (3). Hence
the class of positive multiple discrete sets with almost periodic in the sense
of distributions associate measures coincides with the class of positive
almost periodic multiple discrete sets. But this assertion does not valid
for signed multiple discrete sets.
Theorem 5. There is a signed multiple discrete set such that its associate
measure is almost periodic in the sense of distributions and not almost
periodic in the weak sense.
Proof. Let α(n), n ∈ 2Z \ {0}, be the greatest k ∈ N such that 2k is a
divisor of n. Put
a+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−, where
A+ = {(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.
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56 Perturbations of discrete lattices
Take a function ϕ ∈ C∞ such that suppϕ ⊂ (−1/2, 1/2). Suppose
that τ = 2pk for some p ∈ N, k ∈ Z. If |x− n| ≥ 3/4 for all n ∈ 2Z, then
the 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 for
positive continuous functions ϕ with an arbitrary small diameter of its
support. Hence if A is a signed almost periodic multiple discrete set and
inf{|x− y| : x ∈ s(A+), y ∈ s(A−)} > 0,
then A+ and A− are almost periodic multiple discrete set s as well. But
this is false in the general case.
Theorem 6. There is a signed almost periodic set A such that A+ and
A− are not almost periodic.
Proof. Put
A+ = {(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.
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S. Favorov, Ye. Kolbasina 57
Following the proof of Theorem 5, we can assure that µA is almost
periodic in the sense of distributions. Since the measure µA satisfies
condition (3), we get that A is a signed almost periodic multiple discrete
set.
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 the
distance between the point 2n0 + (α(2n0) + 1)−2 − τ and any point of
A+ is more than 1. We obtain a contradiction. Consequently, A+ is not
almost periodic. Analogously, A− is not almost periodic as well. Theorem
is proved.
Clearly, the measure |µA| = 2µA+ − µA is not almost periodic as well.
Therefore we obtain
Corollary. The positive discrete set {a+n }∪{a−n } does not almost periodic.
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58 Perturbations of discrete lattices
Contact information
S. Favorov Mathematical School, Kharkov National
University, Swobody sq.4, Kharkov, 61077
Ukraine
E-Mail: Sergey.Ju.Favorov@univer.kharkov.ua
URL: www-mechmath.univer.kharkov.ua
/funcan/staff/favorov/index.html
Ye. Kolbasina Mathematical School, Kharkov National
University, Swobody sq.4, Kharkov, 61077
Ukraine
E-Mail: kvr_jenya@mail.ru
URL: -
Received by the editors: 12.02.2010
and in final form ????.
Favorov Sergey and Kolbasina Yevgeniia
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| id | nasplib_isofts_kiev_ua-123456789-154502 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T18:28:35Z |
| publishDate | 2010 |
| publisher | Інститут прикладної математики і механіки НАН України |
| record_format | dspace |
| spelling | Favorov, S. Kolbasina, Y. 2019-06-15T16:12:52Z 2019-06-15T16:12:52Z 2010 Perturbations of discrete lattices and almost periodic sets / S. Favorov, Y. Kolbasina // Algebra and Discrete Mathematics. — 2010. — Vol. 9, № 2. — С. 48–58. — Бібліогр.: 11 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:11K70; 52C07, 52C23. https://nasplib.isofts.kiev.ua/handle/123456789/154502 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. en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics Perturbations of discrete lattices and almost periodic sets Article published earlier |
| spellingShingle | Perturbations of discrete lattices and almost periodic sets Favorov, S. Kolbasina, Y. |
| title | Perturbations of discrete lattices and almost periodic sets |
| title_full | Perturbations of discrete lattices and almost periodic sets |
| title_fullStr | Perturbations of discrete lattices and almost periodic sets |
| title_full_unstemmed | Perturbations of discrete lattices and almost periodic sets |
| title_short | Perturbations of discrete lattices and almost periodic sets |
| title_sort | perturbations of discrete lattices and almost periodic sets |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/154502 |
| work_keys_str_mv | AT favorovs perturbationsofdiscretelatticesandalmostperiodicsets AT kolbasinay perturbationsofdiscretelatticesandalmostperiodicsets |