DEMONSTRATIO MATHEMATICA
Vol. XLII
No 4
2009
Roman Wituła, Damian Słota
THE CONVERGENCE CLASSES OF
DIVERGENT PERMUTATIONS
Abstract. The scope of the paper is the analysis of the convergence classes of different
permutations of N. In this paper we prove that with any divergent permutation p it could
be associated a family D(p) of divergent permutations having the power of the continuum
and such that the convergence class of p is a proper subset of the convergence class of q
for every q ∈ D(p). Also, the convergence classes of the countable families of divergent
permutations are discussed here.
1. Introduction
Let us denote by P the family of all permutations of N.
P
Given a permutation p ∈ P its convergence class,
denoted by (p),
P
is defined to bePthe family of all convergent series
an of real terms such
that the series
ap(n) is also convergent.
We say that two permutations p, q ∈ P are incomparable if the following conditions hold true:
X
X
X
X
(p) \
(q) 6= ∅ and
(q) \
(p) 6= ∅.
A permutation p ∈ P is called
a
P a divergent permutation if there exists
P
conditionally convergent series an of real terms such that the series ap(n)
is divergent. The family of all divergent permutations is denoted by D. We
say that p is a convergent permutation if p ∈ C := P \ D. Thus p ∈P
P is
a convergent permutation
if
for
every
conditionally
convergent
series
an
P
of real terms the series
ap(n) is also convergent.
We introduce the notation AB for the subset of P defined by the following
relation:
p ∈ AB if and only if p ∈ A and p−1 ∈ B
for every A, B ∈ {C, D}. We note that all these families AB are nonempty.
2000 Mathematics Subject Classification: 40A05, 05A99.
Key words and phrases: divergent permutation, convergence class.
788
R. Wituła, D. Słota
In this paper we prove that with any divergent permutation p it could
be associated a family D(p) of P
divergent permutations having
P the power of
the continuum and such that
(p) is a proper subset of
(q) for every
q ∈ D(p). Moreover, it may be assumed that any two different elements of
D(p) are incomparable and, additionally, that D(p) is a subset of one of the
following three sets: either DC or DD or Ω whenever p also belongs to this
set. For the definition of the family Ω we refer to the Section 5. Moreover,
an example of the permutations p, q ∈ D such that there is no divergent
permutation σ with the property
X
X
X
(p) ∪
(q) ⊆
(σ)
is presented in Section 4.
2. Notations and terminology
Let A and B be infinite subsets of N. By D(A, B) and by C(A, B) we
denote two subsets of P defined as follows:
D(A, B) = {p ∈ P : p(A) = B and the composition ψB ◦ (p |A) ◦ φA
is a divergent permutation},
and
C(A, B) = {p ∈ P : p(A) = B and the composition ψB ◦ (p |A) ◦ φA
is a convergent permutation},
where φA is the increasing bijection of N onto A, p |A is the restriction to
the A of p, and ψB is the increasing bijection of B onto N. To simplify the
notation, we write p ∈ U(A) instead of p ∈ U(A, p(A)) where U ∈ {C, D}
and A is an infinite subset of N. We observe that U(N, N) = U for every
U ∈ {C, D}.
Let B be a nonempty subset of N. Then a subset I of B is said to be
an interval of B either if I = ∅ or when it can be expressed in the form
I = B ∩ {n, n + 1, . . . , n + m − 1} for some n, m ∈ N. For abbreviation,
we write "I is an interval" instead of "I is an interval of N". Only intervals
of the subsets of N are discussed in this paper. We will use the symbols
[n, m], [n, m), (n, m] and (n, m) with n, m ∈ N, n ≤ m, to denote the
intervals {n, n + 1, . . . , m}, {n, n + 1, . . . , m − 1}, {n + 1, n + 2, . . . , m} and
{n + 1, n + 2, . . . , m − 1} respectively.
We say A ⊂ N is a union of n (or at least n or at most n) mutually
separated intervals
(abbrev.: MSIs) if there exists a family J of nonempty
S
intervals with J = A, card J = n (or card J ≥ n or card J ≤ n, resp.) and
such that dist(I, J) ≥ 2 for any two different members I and J of J.
The convergence classes of divergent permutations
789
We say that two sequences {xi : i = 1, 2, . . . , n} and {yi : i = 1, 2, . . . , n}
of positive integers are spliced when they are both one-one, have no common
values and satisfy one of the following conditions: either
xp(i) < yq(i) < xp(i+1) < yq(n) for every i = 1, 2, . . . , n − 1,
or
yq(i) < xp(i) < yq(i+1) < xp(n) for every i = 1, 2, . . . , n − 1.
Here p and q denote the permutations of the set {1, 2, . . . , n} chosen in such
a way that the sequences {xp(i) : i = 1, 2, . . . , n} and {yq(i) : i = 1, 2, . . . , n}
are both increasing. In other words, two finite sequences x and y are spliced
if they have the same cardinality, are both one-one, have no common values
and if the increasing sequence in which the elements of x and y alternate
could be created from all elements of the sequences x and y.
A family S of increasing sequences of positive integers having the power of
the continuum is said to be a Sierpiński’s family if any two different members
{an } and {bn } of S are almost disjoint. This means that the sets of values of
the sequences {an } and {bn } have only a finite number of common elements.
We use the symbol c(p|A), where p ∈ C and A is a nonempty subset of
N, to denote the following positive integer:
c(p |A) = max{k ∈ N : there exists an interval I of the set A
such that the p(I) is a union of k MSIs of the set p(A) }.
For abbreviation, we write c(p) instead of c(p | N) for every p ∈ C.
We will denote inclusion by ⊆. The use of the sign ⊂ will be reserved for
cases when the subset is proper.
We shall write K < L for any two subsets K and L of N whenever K = ∅
or K and L are nonempty and k < l for any k ∈ K and l ∈ L.
In this paper we will often identify aPgiven sequence with its set of values.
Moreover, we will consider only series
an of real terms.
3. Countable families of divergent permutations
In this section we consider some basic properties of convergence classes of
a given countable families of divergent permutations. First we remark that
the convergence class of any divergent permutation is wide in the meaning that the intersection of any countable family of the convergence classes
of divergent permutations is nonempty. This is expressed in the following
theorem, which is taken from [8].
Theorem 3.1. For each sequence {pnP
} of divergent
P permutations there
exist two conditionally convergent series
an and
bn of real terms such
790
R. Wituła, D. Słota
that the series
∞
P
apn (k) is convergent to zero for every n ∈ N and the set of
k=1
the limit points of the series
∞
P
bpn (k) is equal to R for every n ∈ N.
k=1
Corollary
T P 3.2. For each sequence {pn } of divergent permutations we
have
(pn ) 6= ∅.
n∈N
Theorem 3.3. Let pn , qn ∈ P, n ∈ N. If there exists an infinite subset A
−1
of N such that pn ∈ C(p−1
n (A)) and qn ∈ D(qn (A)) for every positive integer
n then
∞ X
∞ X
\
[
(pn ) \
(qn ) 6= ∅.
n=1
n=1
Proof. Let for every n ∈ N, sn and σn be the increasing mappings of N onto
−1
the sets p−1
n (A) and qn (A), respectively, and let δ be the increasing mapping
of A onto N. Then, by the hypothesis, the permutation φn defined by φn =
δqn σn is divergent for each n ∈ N. Hence, by Theorem 3.1, there exists a
∞
P
P
bφn (k) ,
conditionally convergent series bn such that all series of the form
k=1
∞
P
bψn (k) , n ∈ N, are certainly
k=1
ψn := δpn sn , n ∈ N, are convergent.
n ∈ N, are divergent. Then the series
convergent because all permutation
Let {a(n)} be the
P increasing sequence of all members of the set
define a new series
dn by setting da(n) = bn for every n ∈ N and
for the remaining indices n ∈ N.
P
It can be readily checked that the series
dn and any of its pn
∞
P
dpn (k) , n ∈ N, are convergent. This
rangements i.e. the series
A. We
dn = 0
– rearfollows
k=1
immediately from the relation
X
X
X
X
bψn (j) ,
bδpn (i) =
dpn (i) =
dpn (i) =
i∈I
i∈B
j∈J
i∈B
which holds for every nonempty interval I, where B = I ∩ p−1
n (A) and
−1 (A)), max s−1 (I ∩ p−1 (A))].
J = [min s−1
(I
∩
p
n
n
n
n
∞
P
dqn (k) , n ∈ N, are diverOn the other hand, all series of the form
k=1
gent. This assertion follows from the following relation which holds for every
nonempty interval J:
X
X
X
X
bφn (j) ,
bδqn (i) =
dqn (i) =
dqn (i) =
i∈I
i∈B
i∈B
j∈J
The convergence classes of divergent permutations
791
where B = I ∩ qn−1 (A) and I = [min σn (J), max σn (J)]. The proof is completed.
Corollary 3.4. Let p, q ∈ P. If there exists an infinite subset A of N
such that p ∈ C(p−1 (A)) and q ∈ D(q −1 (A)) then
X
X
(p) \
(q) 6= ∅.
P
P
Corollary 3.5. Let p, q ∈ P. If
(p) ⊆
(q) then for any infinite
subset A of N the following implication proves true:
if p ∈ C(p−1 (A)) then q ∈ C(q −1 (A)).
P
P
Corollary 3.6. Let p, q ∈ P. If
(p) =
(q) then for any infinite
subset A of N the following relation holds:
p ∈ C(p−1 (A))
if and only if
q ∈ C(q −1 (A)).
Let pi ∈ P and let Ai be an infinite subset of N for every i = 1, 2, . . . , n.
Assume that the sets A1 , A2 , . . . , An , form a partition of N as well as the
sets p1 (A1 ), p2 (A2 ), . . . , pn (An ), do.
Moreover, assume that the action of any permutation pi is concentrated
on the set Ai in the sense that all permutations qi , i = 1, 2, . . . , n, and the
permutation φ are convergent. Here
(
pi (k) for k ∈ N \ Ai ,
qi (k) :=
γi (k) for k ∈ Ai ,
for every i = 1, . . . , n and
φ(k) := γi (k),
for k ∈ Ai and i = 1, . . . , n, where γi is the increasing bijection of the set Ai
onto the set pi (Ai ) for every i = 1, . . . , n. Then there is a following relation
between the convergence classes of permutations pi , i = 1, . . . , n, and the
convergence class of the permutation σ defined by σ(k) = pi (k) for k ∈ Ai
and i = 1, . . . , n.
n P
T
P
(pi ) ⊆ (σ).
Lemma 3.7. We have
i=1
P
Proof. Notice that for any series
ak and for any nonempty interval I the
following equality is satisfied:
n X
n X
X
X
X
X
aφ(k) −
aqi (k) ,
aσ(k) =
api (k) +
k∈I
i=1 k∈I
k∈I
i=1 k∈I
which, by the hypothesis, implies the desired inclusion.
Remark 3.8. There exists the possibility of replacing the weak inequality
sign by the equality sign in the assertion of Lemma 3.7. Indeed, assume
792
R. Wituła, D. Słota
that there exists a positive integer t such that for any interval I there exists a family of pairwise disjoint subintervals I1 , I2 , . . . , Ik of I with k ≤ t
satisfying the following two conditions:
(1) any interval Ii , i = 1, . . . , k, is simultaneously a subset of some set Aj
with j = j(i) ∈ {1, . . . , n},
k
S
Ii ) is a union of at most t MSIs.
(2) the set σ(I \
i=1
Then we have
n P
T
P
(pi ) = (σ).
i=1
The following example shows that the inclusion in the assertion of Lemma 3.7 could be made strictly for every positive integer n. Here we consider
only the case n = 2; the generalization to any n ∈ N is rather straightforward
(and will be omitted here).
Example 3.9. Put
p1 (an + 2i − 1) =
(
an + 2i − 1
for i = 1, 2, . . . , n,
an + 2(i − n − 1) for i = n + 1, n + 2, . . . , 2n,
p1 (an + 2i) = an + 2n + i for i = 0, 1, . . . , 2n − 1,
p2 (an + 2i) = 2n + p1 (an + 2i + 1) for i = 0, 1, . . . , 2n − 1,
and
p2 (an + 2i − 1) = an + i − 1 for i = 1, 2, . . . , 2n,
for
S every positive integer n, where an = 2n(n − 1) + 1. Moreover, set A1 =
{an + 2i − 1 : i = 1, 2, . . . , 2n} and A2 = N \ A1 . Then the hypotheses
n∈N
of Lemma 3.7 are satisfied, which implies the following inclusion:
X
X
X
(3.1)
(p1 ) ∩
(p2 ) ⊆
(σ),
where σ(k) := pi (k) for k ∈ Ai , i = 1, 2.
Now define


−n−1 for k = an + 2i and for k = an + 2n + 2i + 1




where i = 0, 1, . . . , n − 1,
bk =
−1

n
for k = an + 2i − 1 and for k = an + 2n + 2(i − 1)




where i = 1, 2, . . . , n,
P
for n ∈ N. It is clear that the series
bk is convergent. Furthermore,
The convergence classes of divergent permutations
793
a noncomplicated verification shows that
2n
X
bp1 (an +i) = 1 + 0.5[(−1)n − 1]n−1 ,
i=1
2n−1
X
bp2 (an +2n+i) = 1 + 0.5[(−1)n−1 + 1]n−1 ,
i=0
4n−1
X
i=0
bσ(an +i) = 0,
and
X
bσ(i) ≤ 2n−1
i∈I
for any subinterval I of the interval [an , an+1P
) and for every
P n ∈ N. From
the above relations we deduce that the
bp1 (n) and
bp2 (n) are both
P series
divergent but, however, the series
bσ(n) is convergent. Accordingly, by
(3.1), we get
X
X
X
(p1 ) ∩
(p2 ) ⊂
(σ),
which is our claim.
4. Extending of convergence classes
The convergence class of any divergent permutation can be extended
to the convergence class of some other divergent permutation. What is
more, for any divergent permutation p there exists a family D(p) of pairwise
incomparable divergent permutations having the continuum cardinality and
such that the convergence class of any permutation q ∈ D(p) is bigger than
the convergence class of p. This is the main result of this section.
We begin with three auxiliary lemmas. First lemma is well known and
comes from P. Erdős, the two other ones come from paper [10].
Lemma 4.1. Let ak , k = 1, 2, . . . , n, be a 1 − 1 sequence of real numbers.
Then rs ≥ n where r denotes the length of the longest decreasing subsequence
of {ak } and s stands for the length of the longest increasing subsequence of
{ak }.
Lemma 4.2. Let p ∈ P. Suppose that for some interval I and for some
positive integer k the set p(I) is a union of at least (4k 4 + 1) MSIs. Then
there exists an increasing sequence X = {xn : n = 1, 2, . . . , 2k} of positive
integers satisfying the following conditions:
(1) X ⊂ p−1 ([min p(I), max p(I)]),
(2) one of the two following relations holds true:
either {xn : n = 1, 2, . . . , k} ⊆ I and xk+1 > I
or xk < I and {xn : n = k + 1, k + 2, . . . , 2k} ⊆ I,
794
R. Wituła, D. Słota
(3) the sequences {p(xn ) : n = 1, 2, . . . , k} and {p(xn ) : n = k + 1, k +
2, . . . , 2k} are both strictly monotonic and simultaneously, are spliced.
Corollary 4.3. Let p ∈ D. Then for any r, s ∈ N there exists an increasing sequence S = {zn : n = 1, 2, . . . , 2r} of positive integers such that
(1) S ∪ p(S) > s,
(2) the sequences {p(zn ) : n = 1, 2, . . . , r} and {p(zn ) : n = r + 1, r +
2, . . . , 2r} are both strictly monotonic and are spliced.
Lemma 4.4. Let p, q ∈ P. Assume that there exist increasing sequences
(r)
Xr = {xn : n = 1, 2, . . . , 2r}, r ∈ N of positive integers such that for every
r∈N
(1) Xr ∪ p(Xr ) ∪ q −1 p(Xr ) < Xr+1 ∪ p(Xr+1 ) ∪ q −1 p(Xr+1 ),
(1)
(r)
(2)
(1)
(2) the sequences Xr and Xr are spliced where Xr := {p(xn ) : n =
(r)
(2)
1, 2, . . . , r} and Xr := {p(xn ) : n = r + 1, r + 2, . . . , 2r} and, at the
same time,
(r)
(r)
(3) the sequences {q −1 p(xn ) : n = 1, 2, . . . , r} and {q −1 p(xn ) : n = r +
1, r + 2, . . . , 2r} are spliced.
P
P
Then we have (q) \ (p) 6= ∅.
Proof. Put
an =
(
(1)
r−1 for n ∈ Xr ,
(2)
−r−1 for n ∈ Xr ,
for every r ∈ N. For the remaining indices n ∈ N we set an = 0.
P
Obviously, by the assumptions (1) and (2), the series
an is convergent.
By the condition (3) the following two relations hold true for every r ∈ N:
X
(4.1)
aq(n) ≤ r−1
n∈I
for any subinterval I of the interval Jr := [min q −1 p(Xr ), max q −1 p(Xr )] and
X
aq(n) = 0.
(4.2)
n∈Jr
P
By the condition (1) this means that the series
aqn is also convergent. On
the other hand, we have
X
X
an = 1
ap(n) =
(r)
(r)
n∈[x1 ,xr ]
(1)
n∈Xr
for every r ∈P
N which
P yields the divergence of the series
the relation (q) \ (p) 6= ∅ holds as claimed.
P
ap(n) . Therefore
The convergence classes of divergent permutations
795
Theorem 4.5. For every permutation p ∈ D there exists a family Φ(p) of
divergent permutations such that
(1) P
card Φ(p)P= c,
(2)
(p) ⊂ (q) for every permutation q ∈ Φ(p),
(3) any two different permutations q1 , q2 ∈ Φ(p) are incomparable.
Proof. Fix p ∈ D. By Corollary 4.3 we can choose a countable family of
(r)
increasing sequences Sr = {zn : n = 1, 2, . . . , 2r}, r ∈ N, of positive integers
such that for every r ∈ N
Sr ∪ p(Sr ) < Sr+1 ∪ p(Sr+1 ),
(4.3)
(1)
Sr
(2)
and Sr
and the sequences
ously are spliced, where
(1)
(r)
(2)
(r)
are both strictly monotonic and simultane-
Sr := {p(zn ) : n = 1, 2, . . . , r},
(4.4)
Sr := {p(zn ) : n = r + 1, r + 2, . . . , 2r}.
We shall use the following notation:
(r)
z0 = min{Sr ∪ p(Sr )},
(r)
Ii
(r)
(r)
= [zi , zi+1 ),
(r)
(r)
(r)
z2r+1 = max{Sr ∪ p(Sr )},
i = 0, 1, . . . , 2r − 1,
(r)
I2r = [z2r , z2r+1 ], for every r ∈ N.
(r)
It is obvious that the intervals Ii , i = 0, 1, . . . , 2r, form a partition of the
(r) (r)
interval Kr := [z0 , z2r+1 ] and, by (4.3), we have
Kr < Kr+1 r ∈ N.
(4.5)
(r)
Notice that it may happen that some of the sets I0 , r ∈ N, are empty.
Our next goal is to define a new partition of the interval Kr having the
(r)
form {Ji : i = 0, 1, . . . , 2r} for every r ∈ N. The desired partition of the
interval Kr will be uniquely determined by the conditions:
(r)
J0
(r)
(r)
= I0 < Ji
(r)
< Ji+1 ,
for i = 1, 2, . . . , 2r − 1, and
(r)
(r)
card J2i−1 = card Ii
and
(r)
(r)
card J2i = card Ii+r ,
for i = 1, 2, . . . , r.
(r)
(r)
We will denote by γ2i−1 the increasing mapping of the interval J2i−1 onto
(r)
(r)
(r)
the interval Ii and by γ2i the increasing mapping of the interval J2i onto
(r)
the interval Ii+r , for every i = 1, 2, . . . , r and for every r ∈ N. Moreover, let
796
R. Wituła, D. Słota
(r)
(r)
(r)
γ0 be the identity function of the interval I0 whenever I0 6= ∅. In the
(r)
(r)
case when I0 = ∅ we define γ0 as the empty function.
Let S denote a Sierpiński’s family of increasing sequences of positive integers. With each sequence s ∈ S, s = {s(n)}, we associate some permutation
qs acting as follows:
(s(r))
qs (n) = pγi
(n),
(s(r))
for n ∈ Ji
, i = 0, 1, . . . , 2s(r) and r ∈ N. For the remaining positive
integers n we set qs (n) = p(n).
Put D(p)i
= {qs : s ∈ S}. By (4.4) and by the definition of qs the set
h
(r) (r)
is a union of at least r MSIs for every r ∈ N \ s. Since the
qs z1 , zr
set N \ s is infinite we conclude that the permutation qs is divergent for each
s ∈ S. Hence D(p) ⊂ D.
Fix an s ∈ S, s = {s(n)}. Then, from the definition of qs we deduce that for any n ∈ N and for any subinterval U of the interval Ks(n)
h
(s(n)) (s(n))
, zs(n)+1 and
there exist two subintervals V1 and V2 of the intervals z0
i
h
(s(n))
(s(n))
zs(n)+1 , z2s(n)+1 respectively, such that
qs (U ) = p(V1 ) ∪ p(V2 ).
Simultaneously we have
qs (Kr ) = p(Kr )
for every r ∈ N, and
for all positive integers n ∈
qs (n) = p(n)
S
Ks(n) .
N\
n∈N
From the last three relations
and
P
P from (4.5) it can be easily concluded
that the weak inclusion (p) ⊆ (qs ) holds true. Furthermore, if we set
Xr = Ss(r) , r ∈ N, then the hypotheses of Lemma
P 4.4 are
Pfulfilled with qs
instead of the permutation q, and consequently
(q
)
\
(p) 6= ∅. Hence
s
P
P
and from the previous relation we get (p) ⊂ (qs ).
Take s, t ∈ S, s 6= t, s = {s(n)}, t = {t(n)}. Then for almost all indices
n ∈ N we have
(4.6)
qs (i) = p(i),
i ∈ Kt(n) ,
qt (i) = p(i),
i ∈ Ks(n) .
and
(4.7)
The convergence classes of divergent permutations
Set
an =
(
bn =
(
797
(1)
r−1 for n ∈ Ss(r) ,
(2)
−r−1 for n ∈ Ss(r) ,
and
(1)
r−1 for n ∈ St(r) ,
(2)
−r−1 for n ∈ St(r) ,
for every r ∈ N. Immediately from
of the permutations qs
P
P the definitions
convergent.
bqt(n) are bothP
and qt it follows that the series
aqs(n) and
At
the
same
time,
from
(4.6)
and
(4.7)
we
get
that
the
series
aqt(n) and
P
bqs(n) are both divergent. This means that the permutations qs and qt are
incomparable, and hence the proof is completed.
Remark 4.6. If p ∈ DD then we may assume that for every r ∈ N there
exists an interval I such that
(r)
(r+1)
z2r+1 < I ∪ p−1 (I) < z0
and the set p−1 (I) is a union of at least r MSIs. Then from the definition
of qs , we see that each permutation qs , s ∈ S, belongs to DD. Thus D(p) ⊂
DD.
Remark 4.7. Let R be the set of all positive integers r for which only one
(2)
(1)
of the sequences Sr or Sr is increasing. Suppose that the definitions of
(r)
(r)
the intervals Ji , i = 1, 2, . . . , 2r − 1, and the mappings γ2i , i = 1, 2, . . . , r,
(r)
are replaced for every r ∈ R by the following ones. The intervals Ji ⊆ K
are uniquely determined by the conditions:
(r)
J0
(r)
(r)
= I0 < Ji
(r)
= Ji+1
for i = 1, 2, . . . , 2r − 1, and
(r)
(r)
card J2i−1 = card Ii
and
(r)
(r)
card J2i = card I2r−i+1 ,
(r)
for i = 1, 2, . . . , r. Next γ2i is defined to be the increasing mapping of the
(r)
(r)
interval J2i onto the interval I2r−i+1 , for every i = 1, 2, . . . , r.
Take s, t ∈ S, s 6= t, s = {s(n)}, t = {t(n)}. Then it can be easily
concluded that
[
[
qs ∈ C qs−1
p(Ss(n) )
and qt ∈ D qt−1
p(Ss(n) ) ,
n∈N
n∈N
and
[
[
qs ∈ D qs−1
p(St(n) )
and qt ∈ C qt−1
p(St(n) ) .
n∈N
n∈N
798
R. Wituła, D. Słota
This implies, by Corollary 3.4, that the permutations qs and qt are incomparable.
Example 4.8. Let {In } be an increasing sequence of intervals which form
a partition of N and such that card In → ∞ as n → ∞.
Let p, q ∈ D be permutations such that the restriction to the set E of
p is the identity function of E and the restriction to O of q is the identity
function of O, where
[
[
E :=
I2n and O :=
I2n−1 .
n∈N
n∈N
Moreover, assume that
(4.8)
p(In ) = q(In ) = In ,
n ∈ N.
Then, by Remark 3.8, the following equality is fulfilled:
X
X
X
(p) ∩
(q) =
(σ),
where σ(i)
P := p(i)
P for i ∈ O and σ(i) := q(i) for i ∈ E. On the other hand,
the set (p) ∪ (q) is not a subset of the convergence class of any divergent
permutation since the following lemma holds true:
P
P
P
Lemma 4.9. If σ ∈ P and (p) ∪ (q) ⊆ (σ) then σ ∈ C.
Proof. Suppose,Pcontrary
P to ourPclaim, that there exists a permutation
σ ∈ D such that (p) ∪ (q) ⊆ (σ).
Since p ∈ C(p−1 (E)) and q ∈ C(q −1 (O)), from Corollary 3.5 we get
σ ∈ C(σ −1 (E)) and σ ∈ C(σ −1 (O)). The following positive integer k is
therefore well defined:
k = max c σ | σ −1 (E) , c σ | σ −1 (O) .
Now we choose a sequence {Jn } of intervals such that for every n ∈ N
(4.9)
the set σ(Jn ) is a union of at least 2(n + k) MSIs
and
(4.10)
σ(Jn ) < σ(Jn+1 ).
o
n
o
(n)
Let
: i = 1, 2, . . . , vn and Ki : i = 1, 2, . . . , vn − 1 be the increasing MSIs-partitions of the sets
n
(4.11)
(n)
Li
σ(Jn ) and
[min σ(Jn ), max σ(Jn )] \ σ(Jn ),
respectively, for every n ∈ N. There is no loss of generality in assuming that
(4.12)
λn := card {i : Lni ∩ E 6= ∅} ≥ card {i : Lni ∩ O 6= ∅}
for every n ∈ N. By (4.9) we get
(4.13)
λn ≥ n + k,
n ∈ N.
The convergence classes of divergent permutations
799
Since the set Jn ∩ σ −1 (E) is an interval of the set σ −1 (E) it follows from the
definition of k that the set σ(Jn ) ∩ E is a union of at most k MSIs of E.
On the other hand, if a subset I of the set σ(Jn ) ∩ E is simultaneously an
(n)
(n)
interval of E and we have I ∩ Li 6= ∅ and I ∩ Lj 6= ∅ for some indices
(n)
i, j ∈ N such that i < j ≤ vn , then Ks ⊂ O for every index s such that
(n)
(n)
(n)
Li < Ks < Lj . The two last remarks and the conditions (4.12) and
(4.13) assure us that we can choose increasing sequences {s(i) : i = 1, . . . , n}
and {t(i) : i = 1, . . . , n} of positive integers such that


s(n), t(n) < vn ,


(n)
(n)
(n)
(n)
Ls(i) < Kt(i) < Ls(i+1) < Kt(n) ,
(4.14)



(n)
Ls(i)
∩ E 6= ∅ and
(n)
Kt(i)
i < n,
⊂ O,
for every index i = 1, . . . , n.
Let ψ be a choice function of the following family:
o
n
(n)
(n)
U : either U = Ls(i) ∩ E or U = Kt(i) for some i, n ∈ N, i ≤ n .
Define

(w)
−1


 −w for n ∈ {ψ(Kt(i) ) : w ∈ N and i = 1, . . . , w},
(w)
an =
w−1 for n ∈ {ψ(Ls(i) ∩ E) : w ∈ N and i = 1, . . . , w},



0 for the remaining indices n ∈ N.
We notice that by (4.10) this definition is correct.
From (4.8), (4.10),
P (4.14)
P and from the
P definition of ψ we easily deduce
that all three series
an ,P ap(n) and
aq(n) are convergent to zero. On
the other hand, we have
aσ(i) = 1 for every n ∈ N, which is equivalent
i∈Jn P
to the divergence of the series
aσ(n) . This contradicts our assumption.
5. The family Ω
We denote by Ω the family of all divergent permutations p for which
there exist an increasing sequence {In (p)} of intervals and a positive integer
k(p) with the following properties:
(i) p−1 (In (p)) < p−1 (In+1 (p)),
(ii) any set p−1 (In (p)) is a union of at most k(p) MSIs,
(iii) lim γn (p) = ∞ where
n→∞
γn (p) := max c(p | J) : J is an interval and J ⊆ p−1 (In (p)) ,
for every n ∈ N. We notice that DC ⊂ Ω. Furthermore, if p ∈ DC then we
may assume that k(p) = c(p).
800
R. Wituła, D. Słota
Theorem 5.1. For any permutation p ∈ Ω there
a subset Ω(p)
P exists P
of Ω having the power of the continuum such that (p) ⊂ (φ) for each
permutation φ ∈ Ω(p) and any two different permutations φ and ψ from Ω(p)
are incomparable. Moreover, it can be assumed that
Ω(p) ⊂ DC
and
c(φ−1 ) ≤ 4c(p−1 ) + 1,
f or every
φ ∈ Ω(p),
whenever p ∈ DC.
Proof. Let us fix a permutation p ∈ Ω. Take an increasing sequence {In }
of intervals and a positive integer k satisfying the conditions (i)–(iii) above.
Additionally, by passing to a subsequence if necessary, we may suppose that
(5.1)
In ∪ p−1 (In ) < In+1 ∪ p−1 (In+1 ) for every n ∈ N.
Let S be a Sierpiński’s family of increasing sequences of positive integers
with the following property: for any two different sequences s, t ∈ S, s =
{s(n)}, t = {t(n)} the inequallty t(i) > s(i) for some index i ∈ N implies
that
(5.2)
s(n + 1) > t(n) > s(n) for large enough n ∈ N.
With each sequence s ∈ S, s = {s(n)}, we associate the
S permutation
p−1 Is(n) as
φs of N which transforms the elements of the set I(s) :=
n∈N
follows: φs (i) = p(i) for i ∈ I(s) and such that φs is the increasing map of
the complement of the set I(s) onto the complement of the set p(I(s)).
Put Ω(p) = {φs : s ∈ S}. We remark that for each s ∈ S, s = {s(n)},
the conditions (i)–(iii) are satisfied with the permutation φs instead of p and
with the intervals Is(n) instead of In , n ∈ N. Thus Ω(p) ⊂ Ω.
Set Jn = min In ∪ p−1 (In ) , max In ∪ p−1 (In ) , n ∈ N. Then, from
(5.1) and from the definition of the permutations φs , s ∈ S, we deduce that
for every s ∈ S and n ∈ M the following conditions hold true:
(1) φs (Js(n) ) = Js(n) ,
(2) the restriction to the open interval max Js(n) , min Js(n+1) of φs is either
the identity function or the empty function,
−1
(3) and if p ∈ DC then the set φ−1
s (K) is a union of at most 2c(p ) MSIs,
for any subinterval K of the interval Js(n) .
−1
Hence, we get c(φ−1
s ) ≤ 4c(p ) + 1 for all s ∈ S whenever p ∈ DC. This
yields, by (iii), that Ω(p) ⊂ DC whenever p ∈ DC.
Let s, t ∈ S, s = {s(n)}, t = {t(n)} and s 6= t. We show that the
permutations φs and φt are incomparable.
By the conditions (5.2) and (2) there exists m ∈ N such that the re∞
S
striction to the set
It(n) of φs is the identity function and that the
n=m
The convergence classes of divergent permutations
restriction to the set
∞
S
n=m
801
Is(n) of φt , is also the identity function. On the
other hand, by (iii), for any sufficiently large n ∈ N there exist two intervals Ln ⊂ p−1 (Is(n) ) and Kn ⊂ p−1 (It(n) ) and two increasing sequences of
positive integers
n
o
(n)
an = ai : i = 1, 2, . . . , 2(γs(n) − 1) ⊂ Is(n)
and
such that
and
n
o
(n)
bn = bi : i = 1, 2, . . . , 2(γt(n) − 1) ⊂ It(n)
n
o
(n)
φs (Ln ) ∩ an = a2i : i = 1, 2, . . . , γs(n) − 1
n
o
(n)
φt (Kn ) ∩ bn = b2i : i = 1, 2, . . . , γt(n) − 1
for every n ∈ N. Putting these observations together we can easy conclude
that
∞
∞
[
[
−1
−1
φs ∈ D φs
an
and φt ∈ C φt
an ,
n=w
and
n=w
∞
∞
[
[
−1
b
and
φ
∈
D
φ
φs ∈ C φ−1
b
,
n
t
n
s
t
n=w
n=w
for any w ≥ m chosen such that γs(n) > 1 and γt(n) > 1 for n ≥ w. Hence,
by Corollary 3.4, we obtain
X
X
X
X
(φs ) \
(φt ) 6= ∅ and
(φt ) \
(φs ) 6= ∅.
This means that the permutations
P
P φs and φt are incomparable as required.
Now let the series
an and ap(n) be simultaneously convergent. Then
using the definition of permutations φs , s ∈ S, and the P
conditions (ii),
(5.1), (1) and (2) we deduce that any series of the form
aφs (n) , where
s ∈ S, satisfies
the Cauchy condition i.e. it is also convergent. Hence we get
P
P
(p) ⊆ (φs ) for each index s ∈ S.
Finally, since for each s ∈ S there exist infinite many positive integers n ∈
N such that the P
restriction
Pto the interval In of φs is the identity function, the
strict inclusion (p) ⊂ (φs ) follows immediately from the condition (iii)
and from Corollary 3.4.
Acknowledgment. We are grateful to dr J. Włodarz for his helpful
advice and interesting conversations.
References
802
R. Wituła, D. Słota
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INSTITUTE OF MATHEMATICS
SILESIAN UNIVERSITY OF TECHNOLOGY
Kaszubska 23
GLIWICE 44-100, POLAND
E-mail: [email protected]
Received November 12, 2008; revised version 11, 2008.