Rearrangement of conditionally convergent series on a small set

Abstract

We consider ideals I of subsets of the set of natural numbers such that for every conditionally convergent series n∈ω a n and every r ∈ R there is a permutation π r : ω → ω such that n∈ω a πr (n) = r and n ∈ ω: π r (n) = n ∈ I.We characterize such ideals in terms of extendability to a summable ideal (this answers a question of Wilczyński). Additionally, we consider Sierpiński-like theorems, where one can rearrange only indices with positive a n .

On the $$c_0$$-equivalence and permutations of series

“…Filipów and Szuca [11] answered this problem and they proved that a set ideal I has the above-mentioned property if and only if it cannot be extended to a summable ideal.…”

Ideals on countable sets: a survey with questions

“…A classical theorem of Riemann says that any conditional convergent series of reals numbers can be rearranged to converge to any given real number or to diverge to +∞ or −∞. In other words, if (a n ) n is a conditional convergent series and r ∈ R ∪ {+∞, −∞}, there is a permutation π : N → N such that n a π(n) = r. In [18,34] is considered a property of ideals motivated by Riemann's theorem. Let us say that an ideal I has the property R, if for any conditionally convergent series n a n of real numbers and for any r ∈ R ∪ {+∞, −∞}, there is a permutation π : N → N such that n a π(n) = r and {n ∈ N : π(n) = n} ∈ I.…”

Achievement sets and sum ranges with ideal supports

“…To our best knowledge this is a first paper in which achievement set is considered with respect to an ideal, although ideal-sum ranges have been considered before. In [7] Filipów and Szuca defined an ideally supported sum range SR I (x n ) = { ∞ n=1 x σ(n) : σ ∈ S ∞ , supp(σ) = {n : σ(n) = n} ∈ I} for an ideal I. Filipów and Szuca were interested whether SR I (x n ) = R for any conditionally convergent series ∞ n=1 x n . They characterized ideals I with this property, where a crucial role was played by summable ideals.…”

“…Since the convergence is a basic notion in Analysis, most of them deal with ideal convergence of sequences [2], [17], [22]. The following list of topics and related papers is far from being complete and it gives only a flavor of these matters: ideal convergence of sequences of functions [1]; ideal convergence of series [8], [18]; ideal convergence in measure [16], [20]; ideal versions of combinatorial theorems [6]; ideal versions of the Riemann rearrangement theorem and the Levy-Steinitz theorem [7], [16]; ideal version of the Banach principle [11].…”