Permutations preserving sums of rearranged real series

Cent. Eur. J. Math. • 11(5) • 2013 • 956-965DOI: 10.2478/s11533-012-0156-x

Central European Journal of Mathematics

Permutations preserving sumsof rearranged real series

Research Article

Roman Wituła1∗

1 Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland

Received 25 January 2012; accepted 6 July 2012

Abstract: The aim of this paper is to discuss one of the most interesting and unsolved problems of the real series the-ory: rearrangements that preserve sums of series. Certain hypothesis about combinatorial description of thecorresponding permutations is presented and basic algebraic properties of the family S0, introduced by it, areinvestigated.

MSC: 40A05, 05A99

Keywords: Permutations preserving sums of series • Convergent permutations • Divergent permutations© Versita Sp. z o.o.

1. Introduction

Two basic problems are dominating in the theory of real series rearrangements. The first one concerns a combinatorialdescription of permutations p which preserve convergence of rearranged series, i.e. permutations p of N such that forevery conditionally convergent series ∑∞n=1 an the rearranged series ∑∞

n=1 ap(n) is still convergent. Permutations of suchkind will be hereafter called convergent permutations and the family of all convergent permutations will be denoted by C.The family of all permutations of N will be denoted by P. The second problem concerns a combinatorial description ofpermutations p that preserve sums of rearranged series, i.e. such that for every conditionally convergent series∑∞

n=1 anof real terms, the sums

∑∞n=1 an and

∑∞n=1 ap(n) coincide whenever

∑∞n=1 ap(n) is convergent.Quite simple combinatorial descriptions of permutations p ∈ P (and also of functions f : N→ N) preserving convergenceof rearranged series can be found in Levi [6], Agnew [1], Pleasants [7, 8], Schaefer [9], Garibay–Greenberg–Reséndis–Rivaud [2], Guha [3], Wituła et al. [14, 17]. The papers [18–20] present more profound combinatorial descriptions ofpermutations, they are a start point of this paper.

∗ E-mail: [email protected]

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R. Wituła

Recall that according to Agnew’s characterization, a permutation p is convergent if there exists k = k(p) ∈ N suchthat the set p(I) can be partitioned into at most k non-adjacent intervals of N, for every interval I of N. Wituła’scharacterization of the functions f : N → N preserving convergence of f-rearranged series [11] is in the spirit of theabove Agnew’s characterization: let f : N→ N, then convergence of any real series ∑an implies convergence of ∑af(n)iff there exists a positive integer t = t(f) such that for any interval I of N there exists a partition I1, I2, . . . , Is of I, withs ≤ t, with the following properties:

(i) the set Ik is an interval of N,(ii) the restriction of f to Ik is a one-to-one map,(iii) the set f(Ik ) is a union of at most t MSI,for every k = 1, . . . , s. MSI is an abbreviation of “mutually separated intervals”.Definition 1.1.We say that a nonempty set A ⊂ N is a union of n MSI (or of at most n MSI) if there exists a family I of n (or at mostof n) intervals of natural numbers which form a partition of A and dist(I, J) > 2 for any two different I, J ∈ I.From now on, only the intervals of N (segments of N) will be discussed.Combinatorial descriptions of permutations preserving sums of rearranged series are unknown. The aim of this paperis to formulate certain hypothesis concerning this characterization. The hypothesis is: the subfamily S0 of P, seeDefinition 2.1, is the family of all permutations preserving sums of rearranged series. This hypothesis is based on thedeep combinatorial grounds which are described in the next section. Nevertheless, even if this hypothesis does not hold,it seems the family S0 is the most general subfamily of the family S in the sense of combinatorial description, since itcontains all permutations preserving sums of rearranged series known in the literature.The complement of S will be denoted by J, i.e. J = P \S. The elements of J will be called, after Kronrod [5], essentialpermutations. Essential permutations are, in a way, singular permutations because they possess a certain property (seeTheorem 2.8 and the remark after it) dual in relation to the permutations preserving sums of rearranged series.2. Basic concepts

In this section we introduce basic definitions, we also present some indispensable ideas and fundamental facts in thetopic.Definition 2.1.We say that permutation p ∈ P belongs to S0 if there exists a positive integer k = k(p) such that for every n ∈ Nthere exist finite sets An, Bn ⊂ N satisfying the following conditions:1) p(An) = Bn,2) [1, n] ⊂ An,3) each of the sets An and Bn is a union of at most k MSI.Remark 2.2.Note that S0 ⊂ S. Moreover, S0 contains all permutations and functions preserving convergence of rearranged seriesestablished by Agnew and Wituła, respectively. Slightly generalizing Definition 2.1, one can easily introduce thecorresponding family of functions f : N→ N preserving sums of f-rearranged series (this subject will be investigated ina separate paper).

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Permutations preserving sums of rearranged real series

Additionally, we will distinguish some subfamilies of S0 which determine a connection between S0 and the group Ggenerated by the family C of all convergent permutations. As it was proved by Pleasants [7], G 6= P. Permutationsp ∈ P that are not convergent will be called divergent permutations, with the class of divergent permutations denotedby D = P \ C.Denote by D(k), k ∈ N, the set of all divergent permutations p for which there exists an increasing sequence {rn(p)} ofpositive integers such that the set p−1([1, rn(p)]) is a union of at most k MSI, for every n ∈ N. For any A,B ∈ {C,D}denote by AB the family of all those permutations p ∈ A for which p−1 ∈ B. After Kronrod [5] and the author we calla) elements of CC: two-sided convergent permutations,b) elements of CD: one-sided convergent permutations,c) elements of DC: one-sided divergent permutations,d) elements of DD: two-sided divergent permutations.Moreover, define the families

C1 = CD, Cn+1 = (CD ∪DC) ◦ Cn, D1 = DC, Dn+1 = (CD ∪DC) ◦Dn, n ∈ N.

The symbol ◦ denotes here the composition of subsets of P, i.e.B ◦ A = {(q ◦ p)( · ) = q(p( · )) : q ∈ B, p ∈ A},

for any nonempty subsets A,B of P.We note that if p ∈ CD1 then there exists a divergent series ∑an with real terms such that the series ∑ap(n) isconditionally convergent, but if q ∈ CC then for any series ∑an with real terms both series ∑an and ∑aq(n) are,simultaneously, either convergent or divergent. Furthermore, we haveCn ⊆ Cn+1, (1)Dn ⊆ Dn+1, (2)

Cn ∪Dn ⊆ Cn+1 ∩Dn+1, (3)for every n ∈ N, which follow from the relations [15]

CD ◦ CD = CD (so CD ⊂ CD ◦DC), DC ◦DC = DC (so DC ⊂ DC ◦ CD).Moreover, we get the following more compact relations describing families Cn and Dn:

C2 = DC ◦ CD, C3 = CD ◦DC ◦ CD, . . . , D2 = CD ◦DC, D3 = DC ◦ CD ◦DC, . . .

By means of families Cn and Dn, n ∈ N, one can describe the group G,G = ∞⋃

n=1Cn = ∞⋃n=1Dn.

1 If P is a nonempty countable subset of P then there exists a real series∑an and a permutation q ∈ DD such that

the series∑an and

∑ap(n) are divergent for any p ∈ P, whereas the series

∑aq(n) is convergent to zero. The same

fact holds true for q ∈ CD instead of q ∈ DD (an easy technical proof of these facts will be omitted here).

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R. Wituła

For each nonempty subfamily A of P, by A−1 we denote the subfamily P defined as follows:A−1 = {p−1 : p ∈ A}.

For every permutation p of N we denote by t(p, n), n ∈ N, the number of mutually separated intervals which form apartition of the set p([1, n]) and put t(p) = lim supn→∞ t(p, n). It is obvious that p ∈ P is a convergent permutation ifft(p) <∞.We note also that the following theorem holds true, which is immediate from the definition of S0.Theorem 2.3.If p ∈ P and lim inf

n→∞t(p, n) <∞ then p ∈ S0.

On the other hand, Hu and Wang [4] proved that there exists a sum-preserving permutation p such thatlimn→∞

t(p, n) = limn→∞

t(p−1, n) =∞. (4)One can easily verify that permutation p satisfying (4), constructed in the proof of [4, Theorem 7], belongs to S0. We willnow present theorems concerning basic properties and connections between the families of permutations distinguishedabove.Theorem 2.4.The following equalities and inclusions hold true:(i) S−10 = S0;(ii) C ∪ C−1 ⊂ S0;(iii) CC ◦S0 = S0 ◦ CC = S0;(iv) C ◦S0 = S0 ◦ C−1 = S0;(v) (CD ◦S0) ∪ (S0 ◦DC) ⊂ S0;(vi) S0 ⊂ (DC ◦S0) ∩ (S0 ◦ CD) (compare with (v));(vii) CD ◦S0 ◦DC ⊂ S0 ⊂ DC ◦S0 ◦ CD;(viii) D2 ⊂ S0;(ix) D(k) ◦D(l) = D(k)−1◦D(l) = D(k) ◦D(l)−1 = D(k) ◦S0 = S0 ◦D(k) = D(k)−1◦S0 = S0 ◦D(k)−1 = S0 ◦S0 = P,

for any k, l ∈ N. Moreover,

D(k) ∩DC ( D(k+1) ∩DC, k ∈ N,⋃k∈N

D(k) ∩DC = DC.

Conclusion 2.5.The set S0 is not a group because of the equality S0 ◦S0 = P (see (ix)), even though S0 is closed under the inverseoperation (see (i)).Remark 2.6.Since C2 ∩ J 6= ∅ [16, Example] and by (3),

C2 ∪D2 ⊆ C3 ∩D3,thus, the family D3 includes the essential permutations, i.e. D3 ∩ J 6= ∅. Furthermore, from (vii) and C2 ∩ I 6= ∅ wereceive the relationC2 \D2 6= ∅.The above relation is also proved in [16], but in a completely different way. Moreover, let us notice that it is not clearwhether C2 ∩ J = C2\S is a “big” set, for example, whether it is true that (C2\S) ◦ (C2\S) = P. The inclusions

S0 ⊂ S0 ◦ CD and S0 ⊂ DC ◦S0 ◦ CD are strict.959


One should emphasize the fact that no inclusions between families Cn and Dn, n ∈ N, n > 1, are known. The authorassumes that all inclusions (1)–(3) are proper for each n ∈ N (perhaps, adaptations of some Pleasants’ arguments [7]could be useful for solving this problem).Theorem 2.7.The following equalities are true:

J = D \S, (5)J ◦ J = P, (6)

D(1) ◦ J = J ◦D(1) = P. (7)Equality (6) was proved by Kronrod [5]. Nevertheless, for completeness, we present below a proof of (6), but carried outin a different way. At the end we prove the important theorem announced in [13].Theorem 2.8.Let p ∈ P be an essential permutation. Then, for every α ∈ R and for every nonempty closed interval I ⊂ R ∪ {±∞},there exists a conditionally convergent series

∑bn such that∑

bn = α and σbp(n) = I,

where σbp(n) denotes the set of limit points of the series∑bp(n).

It should be added that the property described in the above theorem is not a typical property any more in S. Forexample, if p is a convergent permutation or, more generally, it satisfies the condition lim infn→∞ t(p, n) < ∞, then thesum of arbitrarily given conditionally convergent series∑an belongs to the set of limit points of the p-rearranged series∑ap(n).

3. Proofs

Proof of Theorem 2.4. Let p ∈ S0 and q ∈ C−1. Let us fix the number k ∈ N and finite sets An, Bn ⊂ N, n ∈ N,such that each of the sets An and Bn is a union of at most k MSI, p(An) = Bn, [1, n] ⊂ An, (8)for every n ∈ N. We note that for every m ∈ N there exists n ∈ N such that conditions (8) hold and, additionally,[1, m] ⊂ Bn which implies (i).Moreover, q−1 ∈ C and the set q−1(An) is a union of at most k · c(q−1) MSI, for each n ∈ N. For every convergentpermutation φ of N we denote here by c(φ) the maximum of the number of MSI’s of φ(I) where the maximum is takenover all bounded intervals I of N. Obviously, the following equality is true:

pq(q−1(An)) = Bn, (9)for every n ∈ N. Similarly, the set q−1(Bn) is a union of at most k · c(q−1) MSI and

q−1p(An) = q−1(Bn), (10)for every n ∈ N. From (9) and (10) it follows that

pq ∈ S0 and q−1p ∈ S0, (11)960

R. Wituła

which impliesS0 ◦ C−1 ⊆ S0 and C ◦S0 ⊆ S0. (12)

Whereas, the equality γpq(q−1(An)) = γ(Bn), n ∈ N, where γ ∈ C, impliesC ◦S0 ◦ C−1 ⊆ S0. (13)

Now, if we suppose that q ∈ CC, then from (11) (or (12)) we getp ∈ (CC ◦S0) ∩ (S0 ◦ CC), i.e. S0 ⊆ (CC ◦S0) ∩ (S0 ◦ CC),

which, together with (12), implies (iii) and (iv). Certainly, from (12) the inclusion (v) follows which implies (vi), whereas,from (13) the first one of the weak inclusions of (vii) follows: CD ◦S0 ◦DC ⊂ S0, which implies the second inclusionof (vii): S0 ⊂ DC ◦S0 ◦ CD. By (ii) and (v) we get D2 ⊆ S0 which is (viii). Since⋃k∈N

D(k) ⊆ S0,

in view of (iii), we getCC ◦

⋃k∈N

D(k) ⊆ CC ◦S0 = S0.Moreover, from (i) we obtain (⋃

k∈N

D(k))−1⊆ S0,

which, in view of (iii), impliesCC ◦

(⋃k∈N

D(k))−1⊆ S0.

Now, let us recall an equality proved by Stoller [10],D(1) ◦D(1) = P. (14)

Since D(1)−1 = D(1) andD(1) ⊆ D(k) ( ⋃

n∈N

D(n) ⊆ S0,for every k ∈ N, one can easily deduce from above (basing on (14)) all equalities of (ix).Proof of Theorem 2.7. (5) Since C ⊂ S, we have J ⊂ D, that is J = D \S.(6) Let p ∈ P. We will construct permutations p1, p2 ∈ I such that p = p2p1. Here and thereafter, for any nonemptyA,B ⊂ N we write that A < B if a < b for each a ∈ A and b ∈ B. Moreover, we say that a sequence {An} of nonemptysubsets of N is increasing if An < An+1, n ∈ N. Consider two increasing sequences {In} and {Jn} of intervals of Nhaving the following properties: card In = card Jn = 2n,

p(⋃

n∈N

In

)∩⋃n∈N

Jn = ∅,⋃n∈N

In ∩ p−1(⋃n∈N

Jn

) = ∅ (15)

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andcomplements U , V and W of the sets: ⋃

n∈N

In ∪ p−1(⋃n∈N

Jn

),⋃

n∈N

(In ∪ Jn) and p(⋃

n∈N

In

)∪⋃n∈N

Jn, respectively, are infinite. (16)

Set an = min In, n ∈ N. Let {αn} be an increasing sequence of all elements of the set ⋃n∈N In, whereas, let {βn : n =2, 3, . . .} be a sequence obtained by composing the successive “triplets” of the forms:a2, a2 + 2, a1 + 1,a3, a3 + 2, a2 + 1,a3 + 4, a3 + 6, a2 + 3,. . . . . . . . .

an, an + 2, an−1 + 1,. . . . . . . . .an + 2(2i− 2), an + 2(2i− 1), an−1 + 2i− 1,. . . . . . . . .an + 2n − 4, an + 2n − 2, an−1 + 2n−1 − 1,

where the subscript i varies between 1 and 2n−2

. . . . . . . . .

over all n ∈ N. Additionally, we put β1 = a1.Let σ be an increasing map of the set ⋃n∈N In onto the set ⋃n∈N Jn. Certainly, by (16), σ can be extended to a bijectionbetween N and N. We putp1(αn) = βn and p2(σ (αn)) = σ (βn), n ∈ N.

We can also extend p1 to the set p−1(⋃n∈N Jn

) and p2 to the set ⋃n∈N In so thatp2p1 = p (17)

(it is possible in view of (15)). By (16), the extensions p1 and p2 onto the sets U and V , respectively, can be madeanyhow, only the relation (17) must be preserved. If we take nowγi =

2−n+1 for i = an + 2j − 2, 1 6 j 6 2n−1, n ∈ N,−2−n+1 for i = an + 2j − 1, 1 6 j 6 2n−1, n ∈ N,0 for other i ∈ N,

then ∑γi = 0, ∑

γp1(i) = 1 and ∑γσ−1(i) = 0, ∑

γp2σ−1(i) = 1,hence p1, p2 ∈ I.(7) Let p ∈ P. Take a permutation q ∈ J and a conditionally convergent series ∑∞

n=1 an such that if rearrangedby permutation q, into the series ∑∞n=1 aq(n), is convergent as well but to a different sum. Let {In} be an increasingsequence of intervals of N, selected in such a way that the following conditions are satisfied:p(In) < p(In+1), min In+1 − max In > 2n,

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R. Wituła

for each n ∈ N. We takeA = ⋃

n∈N

In and B = p(A).Let φ be an increasing map of the set A onto N, while ψ an increasing map of N onto B. Now, define permutationsf ∈ D(1) and g ∈ J so that p = gf . Let

the restriction of f to the interval (max In, n+ max In]is an increasing map of this interval onto the set {2i+ max In}ni=1, (18)the restriction of f to the interval (n+ max In,min In+1)is an increasing map of this interval onto the set (max In,min In+1) \ {2i+ max In}ni=1, (19)

for every n ∈ N. Additionally let us assume that the restriction of permutation g to the set A is equal to the composi-tion ψqφ. Then g(A) = B and f(N\A) = N\A. It remains to takef(a) = g−1(p(a)) (20)

for every a ∈ A and g(k) = p(f−1(k)) for every k ∈ N \ A. By (20), (18) and (19) we havef(In) = In, f

((max In,min In+1)) = (max In,min In+1),and the set f ((max In, n+ max In]) is a union of n MSI, for every n ∈ N, so f ∈ D(1).Furthermore, if we define the terms of series ∑∞

n=1 dn in the following way:db = {aψ−1(b) for b ∈ B,0 for b ∈ N \ B,

then we obtain the series arising from ∑∞n=1 an by adding infinite amount of zeros with, however, unchanged order ofterms in∑∞

n=1 an. In this way the series∑∞n=1 dn is convergent and its sum is equal to the sum of∑∞

n=1 an. Furthermore,the series ∑∞n=1 dg(n) is the modified series ∑∞

n=1 aq(n) formed by completing the latter by infinite amount of zeros andpreserving the order of terms in ∑∞n=1 aq(n). It implies the series ∑∞

n=1 dg(n) is convergent and its sum (= ∑∞n=1 aq(n)) isdifferent than the sum ∑∞

n=1 dn =∑∞n=1 an.We have shown that P = J ◦D(1). Proof of the second equality of (7) can now be reduced to a single line:

P = P−1 = (J ◦D(1))−1 = D(1)−1 ◦ J−1 = D(1) ◦ J.Proof of Theorem 2.8. In view of Theorem 2.3 we obtain limn→∞ t(p, n) = ∞ (see also [13]). Hence, by[13, Theorems 3.1 and 5.2] it is enough to prove that for any α, β, γ ∈ R, β < γ, there exists a conditionally con-vergent series ∑bn such that ∑

bn = α and σbp(n) = [β, γ].For this purpose, choose a conditionally convergent series ∑an so that the series ∑ap(n) is convergent as well, but∑ap(n) 6= ∑

an. Fix α, β, γ ∈ R, β < γ. Since p is a divergent permutation (see [12, Theorems 3.1 and 3.3] and[13, Theorem 3.1]), there exists a conditionally convergent series ∑ cn such that∑

cn = α and σcp(n) = [α − γ − β2 , α + γ − β2].

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Putdn = (α − γ + β2

)(an −

12n ∞∑i=1 ai

)/ ∞∑i=1 (ai−ap(i)),

for every n ∈ N. Then ∑dn = 0 and σdp(n) = γ + β2 − α.

Hence, if we take bn = cn + dn, n ∈ N, we obtain∑

bn = α and σbp(n) = [β, γ],as it was required.Remark 3.1.Proof of [13, Theorem 5.2] for essential permutations can be simplified. Let α, β ∈ R and p ∈ I. From [13, Theorem 4.4]and the proof of Theorem 2.8 there exist a real series ∑ fn and ∑gn such that

∑fn = α and σfp(n) = [α,∞],∑gn = 0 and ∑

gp(n) = β − α.

Then ∑(fn+gn) = α and σ (fp(n) +gp(n)) = [β,∞].Acknowledgements

The author wants to express sincere and warmest thanks to the reviewers for their effort put in deep and detailedinvestigation of this paper which resulted in correction of some failures and errors (especially in Theorem 2.4), essentialsupplements in the considerations and improvement of the text clarity.

References

[1] Agnew R.P., Permutations preserving convergence of series, Proc. Amer. Math. Soc., 1955, 6(4), 563–564[2] Garibay F., Greenberg P., Reséndis L., Rivaud J.J., The geometry of sum-preserving permutations, Pacific J. Math.,1988, 135(2), 313–322[3] Guha U.C., On Levi’s theorem on rearrangement of convergent series, Indian J. Math., 1967, 9, 91–93[4] Hu M.C., Wang J.K., On rearrangements of series, Bull. Inst. Math. Acad. Sinica, 1979, 7(4), 363–376[5] Kronrod A., On permutation of terms of numerical series, Mat. Sbornik, 1946, 18(60), 237–280 (in Russian)[6] Levi F.W., Rearrangement of convergent series, Duke Math. J., 1946, 13, 579–585[7] Pleasants P.A.B., Rearrangements that preserve convergence, J. London Math. Soc., 1977, 15(1), 134–142[8] Pleasants P.A.B., Addendum: “Rearrangements that preserve convergence”, J. Lond. Math. Soc., 1978, 18(3), 576[9] Schaefer P., Sum-preserving rearrangements of infinite series, Amer. Math. Monthly, 1981, 88(1), 33–40[10] Stoller G.S., The convergence-preserving rearrangements of real infinite series, Pacific J. Math., 1977, 73(1), 227–231[11] Wituła R., Convergence-preserving functions, Nieuw Arch. Wisk., 1995, 13(1), 31–35[12] Wituła R., The Riemann theorem and divergent permutations, Colloq. Math., 1995, 69(2), 275–287[13] Wituła R., On the set of limit points of the partial sums of series rearranged by a given divergent permutation,J. Math. Anal. Appl., 2010, 362(2), 542–552[14] Wituła R., On algebraic properties of some subsets of families of convergent and divergent permutations (manuscript)964

R. Wituła

[15] Wituła R., The algebraic properties of the convergent and divergent permutations (manuscript)[16] Wituła R., The family F of permutations of N (manuscript)[17] Wituła R., Słota D., Seweryn R., On Erdös’ theorem for monotonic subsequences, Demonstratio Math., 2007, 40(2),239–259[18] Nash-Williams C.St.J.A., White D.J., An application of network flows to rearrangement of series, J. Lond. Math. Soc.,1999, 59(2), 637–646[19] Nash-Williams C.St.J.A., White D.J., Rearrangement of vector series. I, Math. Proc. Cambridge Philos. Soc., 2001,130(1), 89–109[20] Nash-Williams C.St.J.A., White D.J., Rearrangement of vector series. II, Math. Proc. Cambridge Philos. Soc., 2001,130(1), 111–134

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Permutations preserving sums of rearranged real series