Scattered Subsets of Groups
We define scattered subsets of a group as asymptotic counterparts of the scattered subspaces of a topological space and prove that a subset A of a group G is scattered if and only if A does not contain any piecewise shifted IP -subsets. For an amenable group G and a scattered subspace A of G, we sho...
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| author | Banakh, T. O. Protasov, I. V. Slobodianiuk, S. V. Банах, Т. О. Протасов, І. В. Слободянюк, С. В. |
| author_facet | Banakh, T. O. Protasov, I. V. Slobodianiuk, S. V. Банах, Т. О. Протасов, І. В. Слободянюк, С. В. |
| author_sort | Banakh, T. O. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2019-12-05T09:48:08Z |
| description | We define scattered subsets of a group as asymptotic counterparts of the scattered subspaces of a topological space and prove that a subset A of a group G is scattered if and only if A does not contain any piecewise shifted IP -subsets. For an amenable group G and a scattered subspace A of G, we show that μ(A) = 0 for each left invariant Banach measure μ on G. It is also shown that every infinite group can be split into ℵ0 scattered subsets. |
| first_indexed | 2026-03-24T02:16:25Z |
| format | Article |
| fulltext |
UDC 512.5
T. O. Banakh (Lviv Nat. Ivan Franko Univ.),
I. V. Protasov, S. V. Slobodianiuk (Kyiv Nat. Taras Shevchenko Univ.)
SCATTERED SUBSETS OF GROUPS
РОЗРIДЖЕНI ПIДМНОЖИНИ ГРУП
We define scattered subsets of a group as asymptotic counterparts of the scattered subspaces of a topological space and
prove that a subset A of a group G is scattered if and only if A does not contain any piecewise shifted IP -subsets. For an
amenable group G and a scattered subspace A of G, we show that µ(A) = 0 for each left invariant Banach measure µ on
G. It is shown that every infinite group can be split into ℵ0 scattered subsets.
Розрiдженi пiдмножини групи визначено, як асимптотичнi аналоги розрiджених пiдпросторiв топологiчного прос-
тору. Доведено, що пiдмножина A групи G є розрiдженою тодi i тiльки тодi, коли A не мiстить кусково-зсунутих
IP -пiдмножин. Показано, що для аменабельної групи G та розрiдженого пiдпростору A групи G рiвнiсть µ(A) = 0
виконується для кожної лiвої iнварiантної банахової мiри µ на G. Встановлено, що кожну нескiнченну групу можна
розбити на ℵ0 розрiджених пiдмножин.
1. Introduction. Given a discrete space X, we take the points of βX, the Stone – Čech compactifi-
cation of X, to be the ultrafilters on X, with the points of X identified with the principal ultrafilters
on X . The topology on βX can be defined by stating that the sets of the form A = {p ∈ βX :
A ∈ p}, where A is a subset of X, form a base for the open sets. We note that the sets of this form are
clopen and that, for any p ∈ βX and A ⊆ X, A ∈ p if and only if p ∈ A. For any A ⊆ X we denote
A∗ = A ∩ G∗, where G∗ = βG \ G. The universal property of βG states that every mapping f :
X → Y, where Y is a compact Hausdorff space, can be extended to the continuous mapping
fβ : βX → Y .
Now let G be a discrete group. Using the universal property of βG, we can extend the group
multiplication from G to βG in two steps. Given g ∈ G, the mapping
x 7→ gx : G→ βG
extends to the continuous mapping
q 7→ gq : βG→ βG.
Then, for each q ∈ βG, we extend the mapping g 7→ gq defined from G into βG to the continuous
mapping
p 7→ pq : βG→ βG.
The product pq of the ultrafilters p, q can also be defined by the rule: given a subset A ⊆ G,
A ∈ pq ↔ {g ∈ G : g−1A ∈ q} ∈ p.
To describe the base for pq, we take any element P ∈ p and, for every x ∈ P, choose some element
Qx ∈ q. Then ∪x∈pxQx ∈ pq, and the family of subsets of this form is a base for the ultrafilter pq.
c© T. O. BANAKH, I. V. PROTASOV, S. V. SLOBODIANIUK, 2015
304 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
SCATTERED SUBSETS OF GROUPS 305
By the construction, the binary operation (p, q) 7→ pq is associative, so βG is a semigroup, and
G∗ is a subsemigroup of βG. For each q ∈ βG, the right shift x 7→ xq is continuous, and the left
shift x 7→ xq is continuous for each g ∈ G.
For the structure of a compact right topological semigroup βG and plenty of its applications to
combinatorics, topological algebra and functional analysis see [1 – 5].
Given a subset A of a group G and an ultrafilter p ∈ G∗, we define a p-companion of A by
∆p(A) = A∗ ∩Gp = {gp : g ∈ G,A ∈ gp},
and say that a subset S of G∗ is an ultracompanion of A if S = ∆p(A) for some p ∈ G∗. For
ultracompanions of subsets of groups and metric spaces see [6, 7].
Clearly, A is finite if and only if ∆p(A) = ∅ for each p ∈ G∗.
We say that a subset A of a group G is
thin if |∆p(A)| ≤ 1 for each p ∈ G∗;
n-thin, n ∈ N if |∆p(A)| ≤ n for each p ∈ G∗;
sparse if each ultracompanion of A is finite;
disparse if each ultracompanion of A is discrete;
scattered if, for each infinite subset Y of A, there is p ∈ Y ∗ such that ∆p(Y ) is finite.
We denote by [G]<ω the family of all finite subsets of G. Given any F ∈ [G]<ω and g ∈ G, we
put
B(g, F ) = Fg ∪ {g}
and, following [8], say that B(g, F ) is a ball of radius F around g. For a subset Y of G, we put
BY (g, F ) = Y ∩ B(g, F ). By [6] (Proposition 4), Y is n-thin if and only if for every F ∈ [G]<ω,
there exists H ∈ [G]<ω such that |BY (y, F )| ≤ n for each y ∈ Y \H . For thin subsets of a group,
their applications and modifications see [9 – 19].
By [6] (Proposition 5) and [20] (Theorems 3 and 10), for a subset A of a group G, the following
statements are equivalent:
(1) A is sparse;
(2) for every infinite subset X of G, there exists finite subset F ⊂ G such that
⋂
g∈F gA is finite;
(3) for every infinite subset Y of A, there exists F ∈ [G]<ω such that, for every H ∈ [G]<ω, we
have
{y ∈ Y : BA(y,H) \BA(y, F ) = ∅} 6= ∅;
(4) A has no subsets asymorphic to the subset W2 = {g ∈ ⊕ωZ2 : supt g ≤ 2} of the group
⊕ωZ2, where supt g is the member of nonzero coordinates of g.
The notion of asymorphisms and coarse equivalence will be defined in the next section. The
sparse sets were introduced in [21] in order to characterise strongly prime ultrafilters in G∗, the
ultrafilters from G∗ \G∗G∗. More on sparse subsets can be find in [10, 11, 16, 22].
In this paper, answering Question 4 from [6], we prove that a subset A of a group G is scattered
if and only if A is disparse, and characterize the scattered subsets in terms of prohibited subsets. We
answer also Question 2 from [6] proving that each scattered subset of an amenable group is absolute
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
306 T. O. BANAKH, I. V. PROTASOV, S. V. SLOBODIANIUK
null. We prove that every infinite group G can be partitioned into ℵ0 scattered subsets. The results
are exposed in Section 2, their proofs in Section 3.
2. Results. Our first statement shows that, from the asymptotic point of view [23], the scattered
subsets of a group can be considered as the counterparts of the scattered subspaces of a topological
space.
Proposition 1. For a subset A of a group G, the following two statements are equivalent:
(i) A is scattered;
(ii) for every infinite subset Y of A, there exists F ∈ [G]<ωsuch that, for every H ∈ [G]<ω, we
have
{y ∈ Y : BY (y,H) \BY (y, F ) = ∅} 6= ∅.
Proposition 2. A subset A of a group G is scattered if and only if, for every countable subgroup
H of G, A ∩H is scattered in H.
Let A be a subset of a group G, K ∈ [G]<ω. A sequence a0, . . . , an in A is called K-chain from
a0 to an if ai+1 ∈ B(ai,K) for each i ∈ {0, . . . , n− 1}. For every a ∈ A, we denote
B�
A(a,K) = {b ∈ A : there is a K-chain from a to b}
and, following [24] (Chapter 3), say that A is cellular (or asymptotically zero-dimentional) if, for
every K ∈ [G]<ω, there exists K ′ ∈ [G]<ω such that, for each a ∈ A,
B�
A(a,K) ⊆ BA(a,K ′).
Now we need some more asymptology (see [24], Chapter 1). Let G, H be groups, X ⊆ G, Y ⊆ H .
A mapping f : X → Y is called a ≺-mapping if, for every F ∈ [G]<ω, there exists K ∈ [G]<ω such
that, for every x ∈ X,
f(BX(x, F )) ⊆ BY (f(x),K).
If f is a bijection such that f and f−1 are ≺-mappings, we say that f is an asymorphism. The subsets
X and Y are called coarse equivalent if there exist asymorphic subsets X ′ ⊆ X and Y ′ ⊆ Y such
that X ⊆ BX(X ′, F ), Y ⊆ BY (Y ′,K) for some F ∈ [G]<ω and K ∈ [H]<ω.
Following [23], we say, that the set Y of G has no asymptotically isolated balls if Y does
not satisfy Proposition 1(ii): for every F ∈ [G]<ω, there exists H ∈ [G]<ω such that BY (y,H) \
BY (y, F ) 6= ∅ for each y ∈ Y .
By [23], a countable cellular subset Y of G with no asymptotically isolated balls is coarsely
equivalent to the group ⊕ωZ2.
Proposition 3. Let X be a countable subset of a group G. If X is not cellular, then X contains
a subset Y coarsely equivalent to ⊕ωZ2.
Let (gn)n<ω be an injective sequence in a group G. The set
{gi1gi2 . . . gin : 0 ≤ i1 < i2 < . . . < in < ω}
is called an IP -set [1, p. 406], the abriviation for “infinite dimensional parallelepiped”.
Given a sequence (bn)n<ω in G, we say that the set
{gi1gi2 . . . ginbin : 0 ≤ i1 < i2 < . . . < in < ω}
is a piecewise shifted IP -set.
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SCATTERED SUBSETS OF GROUPS 307
Theorem 1. For a subset A of a group G, the following statements are equivalent:
(i) A is scattered;
(ii) A is disparse;
(iii) A contains no subsets coarsely equivalent to the group ⊕ωZ2;
(iv) A contains no piecewise shifted IP -sets.
By the equivalence (i) ⇔ (ii) and Propositions 10 and 12 from [6], the family of all scattered
subsets of an infinite group G is a translation invariant ideal in the Boolean algebra of all subsets of
G strictly contained in the ideal of all small subsets.
Now we describe some relationships between the left invariant ideals SpG, ScG of all sparse and
scattered subsets of a group G on one hand, and closed left ideals of the semigroup βG.
Let J be a left invariant ideal in the Boolean algebra PG of all subsets of a group G. We set
Ĵ = {p ∈ βG : G \A ∈ p for each A ∈ J}
and note that Ĵ is a closed left ideal of the semigroup βG. On the other hand, for a closed left ideal
L of βG, we set
Ľ = {A ⊆ G : A /∈ p for each p ∈ L}
and note that Ľ is a left invariant ideal in PG. Moreover, ˇ̂
J = J and ˆ̌L = L.
Clearly, ˆ[G]<ω = G∗ and by Theorem 1,
ˆScG = cl{p ∈ βG : Gp is discrete in βG} =
= cl{p ∈ βG : p = εp for some idempotent ε ∈ G∗}. (1)
Given a left invariant ideal J in PG and following [11], we define a left invariant ideal σ(J) by the
rule: A ∈ σ(J) if and only if ∆p(A) is finite for every p ∈ Ĵ . Equivalently, σ(J) = cl(Ǧ∗Ĵ). Thus,
we have
ˆSpG = cl(G∗G∗).
We say that a left invariant ideal J in PG is sparse-complete if σ(J) = J and denote by σ∗(J) the
intersection of all sparse-complete ideals containing J . Clearly, the sparse-completion σ∗(J) is the
smallest sparse-complete ideal such that J ⊆ σ∗(J). By [11] (Theorem 4(1)), σ∗(J) =
⋃
n∈ω σ
n(J),
where σ0(J) = J and σn+1(J) = σ(σn(J)). We can prove that A ∈ σn([G]<ω) if and only if A has
no subsets asymorphic to Wn = {g ∈ ⊕ωZ2 : supt g ≤ n}.
By [11] (Theorem 4(2)), the ideal SpG is not sparse complete. By (1), the ideal ScG is sparse-
complete. Hence σ∗([G]<ω) ⊆ ScG but σ∗([G]<ω) 6= ScG.
Recall that a subset A of an amenable group G is absolute null if µ(A) = 0 for each left invariant
Banach measure µ onG. For sparse subsets, the following theorem was proved in [10] (Theorem 5.1).
Theorem 2. Every scattered subset A of an amenable group G is absolute null.
Let A be a subset of Z. The upper density d(A) is denoted by
d(A) = lim sup
n→∞
|A ∩ {−n,−n+ 1, . . . , n− 1, n}|
2n+ 1
.
By [25] (Theorem 11.11), if d(A) > 0, then A contains a piecewise shifted IP -set. We note
that Theorem 2 generalizes this statement because there exists a Banach measure µ on Z such that
d(A) = µ(A).
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
308 T. O. BANAKH, I. V. PROTASOV, S. V. SLOBODIANIUK
In connection with Theorem 1, one may ask if it possible to replace piecewise shifted IP -sets to
(left or right) shifted IP -sets. By Theorem 2 and [25] (Theorem 11.6), this is impossible.
By Theorem 1 and Proposition 13 from [6] an infinite group G can be partitioned into ℵ0 scattered
subsets provided that G is embeddable into a direct product of countable groups, in particular if G is
Abelian.
Theorem 3. Every infinite group G can be partitioned into ℵ0 scattered subsets.
We note that Theorem 3 does not hold with sparse subsets in place of scattered subsets [22].
By Theorems 1 and 3, every infinite groups admits a countable partition such that each cell has
no piecewise-shifted IP -sets.
We recall that a free ultrafilter p on a set X is countably complete if, for every countable partition
of X, one cell of the partition is a member of p. In this case |X| should be Ulam-measurable. Let p
be a countably complete ultrafilter on a group G. Applying Theorem 3, we conclude that the orbit
Gp is discrete in G∗.
3. Proofs. Proof of Proposition 1. (i) ⇒ [(ii). We take p ∈ Y ∗ such that ∆p(Y ) is finite, so
∆p(Y ) = Fp for some F ∈ [G]<ω. Given any H ∈ [G]ω, we have hp /∈ ∆p(Y ) for each h ∈ H \F .
Hence hPh ∩ Y = ∅ for some Ph ∈ p. We put P =
⋂
h∈H\P Ph and note that
P ⊆ {y ∈ Y : BY (y,H) \BY (y, F ) = ∅}.
(ii) ⇒ (i). We take an infinite subset Y of A, choose corresponding F ∈ [G]<ω and, for each
H ∈ [G]<ω, denote
PH = {y ∈ Y : BY (y,H) \BY (y, F ) = ∅}.
By (ii), the family {PH : H ∈ [G]<ω} has a finite intersection property and
⋂
H∈[G]<ω PH = ∅.
Hence {PH : H ∈ [G]<ω} is contained in some ultrafilter p ∈ Y ∗. By the choice of p, we have
gp /∈ ∆p(Y ) for each g ∈ G \ (F ∪ {e}), e is the identity of G. It follows that ∆p(Y ) is finite so A
is scattered.
Proof of Proposition 2. Assume that A is not scattered and choose a subset Y of A which does
not satisfy the condition (ii) of Proposition 1. We take an arbitrary a ∈ A and put F0 = {e, a}. Then
we choose inductively a sequence (Fn)n∈ω in [G]<ω such that
(1) FnF−1n ⊂ Fn+1;
(2) BY (y, Fn+1) \BY (y, Fn) 6= ∅ for every y ∈ Y .
After ω steps, we put H =
⋃
n∈ω Fn. By the choice of F0, Y ∩H 6= ∅. By (1), H is a subgroup.
By (2), (Y ∩H) is not scattered in H .
Proof of Proposition 3. Replacing G by by the subgroup generating by X, we assume that G is
countable. We write G as an union of an increasing chain Fn of finite subsets such that F0 = {e},
Fn = F−1n . In view of [23], it suffices to find a cellular subset Y of X with no asymptotically
isolated balls.
Since X is not cellular, there exists F ∈ [G]<ω such that
(1) for every n ∈ N, there is x ∈ X such that
B�
X(x, F ) \BX(x, Fn) 6= ∅.
We assume that G is finitely generated and choose a system of generators K ∈ [G]<ω such that
K = K−1 and F ⊆ K. Then we consider the Cayley graph Γ = Cay(G,K) with the set of vertices
G and the set of edges {{g, h} : g−1h ∈ K}. We endow Γ with the path metric d and say that
a sequence a0, . . . , an ∈ G is a geodesic path if a0, . . . , an is the shortest path from a0 to an, in
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
SCATTERED SUBSETS OF GROUPS 309
particular, d(a0, an) = n. Using (1), for each n ∈ N, we choose a geodesic path Ln of length 3n
such that Ln ⊂ X and
(2) BG(Ln, Fn ∩ BG(Ln+1, Fn+1) = ∅ for every n ∈ N. Let Ln = {an0, . . . , an3n}. For each
i ∈ {0, . . . , 3n}, we take a tercimal decomposition of i and denote by Yn the subset of all ai ∈ L
such that i has no 1-s in its decomposition based on {0, 1, 2} (see [26]). By [2] and the construction
of Yn, the set Y =
⋃
n∈N Yn is cellular and has no asymptotically isolated balls.
Now let G be an arbitrary countable group. We consider a subgroup H of G generated by F and
decomposeG into left cosets byH . IfX meets only finite number of these cosets, thenX is contained
in some finitely generated subgroup of G and we arrive in the previous case. At last, let {Hxn :
n ∈ N} be a decomposition of G into left cosets by H and X meets infinitely many of them.
We endow each Hxn with the structure of a graph Γn naturally isomorphic to the Cayley graph
Cay(H,F ). Then we use (1) to choose an increasing sequence (mn)n∈ω and a sequence (Ln)n∈N
of geodesic paths of length 3n satisfying (2) and such that Ln ⊂ X . For each n ∈ N, we define a
subset Yn of Ln as before, and put Y =
⋃
n∈N Yn. By (2) and the construction of Yn, Y is cellular
and has no asymptotically isolated balls.
Proof of Theorem 1. We follow the tour (i)⇒ (iv) ⇒ (ii)⇒ (iii)⇒ (i).
(i) ⇒ (iv). We prove that a piecewise shifted IP -subset
A = {gi1gi2 . . . ginbin : 0 ≤ i1 < . . . < in < ω}
of G is not scattered. For each m ∈ ω, let
Am = {gi1gi2 . . . ginbin : m < i1 < . . . < in < ω}.
We take an arbitrary p ∈ A∗ and show that ∆p(A) is infinite.
If Am ∈ p for every m ∈ ω, then gnp ∈ A∗ for each n ∈ ω. Otherwise, there exists m ∈ ω such
that
{gmgi1 . . . ginbin : m < i1 < . . . < in < ω} ∈ p.
Then g−1m p ∈ A∗ and we repeat the arguments for g−1m p.
(iv) ⇒ (ii). Assume that A is not disparse and take p ∈ A∗ such that p is not isolated in ∆p(A).
Then p = qp for some q ∈ G∗. The set {x ∈ G∗ : xp = p} is a closed subsemigroup of G∗ and, by
[1] (Theorem 2.5), there is an idempotent r ∈ G∗ such that p = rp. We take R ∈ r and Pg ∈ p,
g ∈ R such that
⋃
g∈R gPg ⊆ A.Since r is an idempotent, by [1] (Theorem 5.8), there is an injective
sequence (gn)n∈ω in G such that
{gi1 . . . gin : 0 ≤ i1 < . . . < in < ω} ⊆ R.
For each n ∈ ω, we pick bn ∈
⋂
{Pg : g = gi1 . . . gin : 0 ≤ i1 < . . . < in < ω} and note that
{gi1 . . . ginbin : 0 ≤ i1 < . . . < in < ω} ⊆ A.
(ii) ⇒ (iii). We assume that A contains a subset coarsely equivalent to the group B = ⊕ωZ2.
Then there exist a subset X of B, H ∈ [B]<ω such that B = H +X, and an injective ≺-mapping f :
X → A. We take an arbitrary idempotent r ∈ B∗, pick h ∈ H such that h+X ∈ r and put p = r−h.
Since r+p = r, we see that p is not isolated in ∆p(X). We denote q = fβ(p). Let b ∈ B, b 6= 0 and
b+ p ∈ X∗. Since f is an injective ≺-mapping, there is g ∈ G \ {e} such that fβ(b+ p) = g+ q. It
follows that q is not isolated in ∆q(A). Hence A is not disparse.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
310 T. O. BANAKH, I. V. PROTASOV, S. V. SLOBODIANIUK
(iii) ⇒ (i). Let X be a countable subset of A. By Proposition 3, X is cellular. By [23], X
satisfies Proposition 1(ii). Hence X is scattered. By Proposition 2, A is scattered.
Proof of Theorem 2. We assume that µ(A) > 0 for some Banach measure µ on G. We use the
arguments from [10, p. 506, 507] to choose a decreasing sequence (An)n∈ω of subsets of G and an
injective sequence (gn)n∈ω in G such that A0 = A, gnAn+1 ⊆ An and µ(An) > 0 for each n ∈ ω.
We pick xn ∈ An+1 and put
X = {gε00 . . . gεnn xn : n ∈ ω , εi ∈ {0, 1}}.
By the construction X is a piecewise shifted IP -sets and X ⊆ A. By Theorem 1, X is not scattered.
Theorem 2 is proved.
To prove Theorem 3, we need some definitions and notations.
Let G be an infinite group with the identity e, κ be an infinite cardinal. A family {Gα : α < κ}
of subgroups of G is called a filtration if the following conditions hold:
(1) G0 = {e} and G =
⋃
α<κ Gα;
(2) Gα ⊂ Gβ for all α < β < κ;
(3)
⋃
α<β Gα = Gβ for each limit ordinal β < κ.
Clearly, a countable group G admits a filtration if and only if G is not finitely generated. Every
uncountable group G of cardinality κ admits a filtration satisfying the additional condition |Gα| < κ
for each α < κ.
Following [27], for each 0 < α < κ, we decompose Gα+1 \ Gα into right cosets by Gα and
choose some system Xα of representatives so Gα+1 \ Gα = GαXα. We take an arbitrary element
g ∈ G\{e} and choose the smallest subgroup Gα with g ∈ Gα. By (3), α = α1 +1 for some ordinal
α1 < κ. Hence, g ∈ Gα+1 \ Gα1 and there exist g1 ∈ Gα1 and xα1 ∈ Xα1 such that g = g1xα1 .
If g1 6= e, we choose the ordinal α2 and elements g2 ∈ Gα2+1 \ Gα2 and xα2 ∈ Xα2 such that
g1 = g2xα2 . Since the set of ordinals {α : α < κ} is well-ordered, after finite number s(g) of steps,
we get the representation
g = xαs(g)
xαs(g)−1
. . . xα2xα1 , xαi ∈ Xαi .
We note that this representation is unique. For n ∈ N, we denote
Dn = {g ∈ G \ {e} : s(g) = n}.
Given any g = xαs(g)
. . . xα2xα1 and m ∈ {1, . . . , s(g)}, we put
g(m) = xαs(g)
. . . xαm , max g = α1.
For a subset P ⊆ Dn and m ∈ {1, . . . , n}, we denote
P (m) = {g(m) : g ∈ P},
maxP = {max g : g ∈ P}.
Let p be an ultrafilter on G such that Dn ∈ p and m ∈ {1, . . . , n}. We denote by max p the
ultrafilter on κ with the base {maxP : P ∈ p, P ⊆ Dn}, and by p(m) the ultrafilter on G with the
base {P (m) : P ∈ p, P ⊆ Dn}.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
SCATTERED SUBSETS OF GROUPS 311
Proof of Theorem 3. We use an induction by the cardinality of G. If G is countable, the
statement is evident because each singleton is scattered. Assume that we have proved the theorem
for all groups of cardinality less than κ, κ > ℵ0 and take an arbitrary group G of cardinality κ. We
fix a filtration {Gα : α < κ} of G such that |Gα| < κ for each α < κ.
For every α < κ, we use the inductive hypothesis to define a mapping χα : Gα+1 \ Gα → N
such that χ−1α (i) is scattered in Gα+1 for every i ∈ N. Then we take g ∈ G \ {e}, g = xαn . . . xα1
and put
χ(g) = (χαn(xαn), χαn−1(xαnxαn−1), . . . , χα1(xαnxαn−1 . . . xα1)).
Thus, we have defined a mapping χ : G\{e} →
⋃
n∈NNn. In view of Theorem 1, it suffices to verify
that χ−1(m) is disparse for eachm = (m1, . . . ,mn) ∈ Nn. We shall prove this statement by induction
on n. Let m = (m1) and let p be an ultrafilter on G such that χ−1(m) ∈ p. We endow κ with the
interval topology and denote λ = lim max p. If p is a principal ultrafilter then, by the definition of χλ,
Gp is discrete. Otherwise, we take P ∈ p such that max p ≤ λ. Then P /∈ gp for each g ∈ G \ {e}.
Suppose that we have proved that χ−1(m′) is disparse for each m′ = (m′1, . . . ,m
′
i), i < n and
let m = (m1, . . . ,mn). We take an arbitrary ultrafilter p on G such that χ−1(m) ∈ p. If max p is
a principal, we use the inductive hypothesis. Otherwise, we denote λ1 = lim max p(1), . . . , λn =
= lim max p(n). We choose k ∈ {1, . . . , n} such that λ1 = . . . = λk, λk+1 < λk. By the inductive
hypothesis, there exists P ∈ p such that P (k + 1) /∈ gp(k + 1) for every g ∈ G \ {e} such that
max g ≤ λk+1. Then we choose Q ∈ p, Q ⊆ Dn such that max q(k+ 1) ≤ λk+1, max q(k) > λk+1
for each q ∈ Q. Then Gp ∩ (P ∩Q) = {p}, so Gp is discrete.
Theorem 3 is proved.
The referee pointed out that, with another definition, the scattered subsets appeared in [28].
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ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
312 T. O. BANAKH, I. V. PROTASOV, S. V. SLOBODIANIUK
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Received 08.12.13
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 3
|
| id | umjimathkievua-article-1983 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:16:25Z |
| publishDate | 2015 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| spelling | umjimathkievua-article-19832019-12-05T09:48:08Z Scattered Subsets of Groups Розріджені підмножини груп Banakh, T. O. Protasov, I. V. Slobodianiuk, S. V. Банах, Т. О. Протасов, І. В. Слободянюк, С. В. We define scattered subsets of a group as asymptotic counterparts of the scattered subspaces of a topological space and prove that a subset A of a group G is scattered if and only if A does not contain any piecewise shifted IP -subsets. For an amenable group G and a scattered subspace A of G, we show that μ(A) = 0 for each left invariant Banach measure μ on G. It is also shown that every infinite group can be split into ℵ0 scattered subsets. Розріджені підмножини групи визначено, як асимптотичні аналоги розраджених підпросторів топологічного простору. Доведено, що підмножина $A$ групи $G$ є розрідженою тоді i тільки тоді, коли $A$ не містить кусково-зсунутих IP-підмножин. Показано, що для аменабельної групи $G$ та розрідженого підпростору $A$ групи $G$ рівність $μ(A) = 0$ виконується для кожної лівої інваріантної банахової міри $μ$ на $G$. Встановлено, що кожну нескінченну групу можна розбити на $ℵ_0$ розріджених підмножин. Institute of Mathematics, NAS of Ukraine 2015-03-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1983 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 3 (2015); 304-312 Український математичний журнал; Том 67 № 3 (2015); 304-312 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1983/985 https://umj.imath.kiev.ua/index.php/umj/article/view/1983/986 Copyright (c) 2015 Banakh T. O.; Protasov I. V.; Slobodianiuk S. V. |
| spellingShingle | Banakh, T. O. Protasov, I. V. Slobodianiuk, S. V. Банах, Т. О. Протасов, І. В. Слободянюк, С. В. Scattered Subsets of Groups |
| title | Scattered Subsets of Groups |
| title_alt | Розріджені підмножини груп |
| title_full | Scattered Subsets of Groups |
| title_fullStr | Scattered Subsets of Groups |
| title_full_unstemmed | Scattered Subsets of Groups |
| title_short | Scattered Subsets of Groups |
| title_sort | scattered subsets of groups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1983 |
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