Qualitative properties of ideal convergent subsequences and rearrangements

The Baire Category of Subsequences and Permutations which preserve Limit Points

“…(L5) =⇒ (L1) Suppose that L x = ∅, otherwise the claim is trivial. Let L be a countable dense subset of L x , so that L ⊆ Λ σ(x) (I, 1 2 ) for each σ ∈S := ∈LS ( , 1 2 ), which is comeager by Corollary 4.4. Fix σ ∈S.…”

“…We identify each subsequence of (x kn ) of x with the function σ ∈ Σ defined by σ(n) = k n for all n ∈ N and, similarly, each rearranged sequence (x π(n) ) with the permutation π ∈ Π, cf. [1,3,29]. We will show that if I is a meager ideal and x is a sequence with values in a separable metric space then the set of subsequences (and permutations) of x which preserve the set of I-cluster points of x is not meager if and only if every ordinary limit point of x is also an I-cluster point of x (Theorem 2.2).…”

Characterizations of ideal cluster points

“…This condition is strictly related to the so-called "property (G)" considered in [3] and to the definition of invariant and thinnable ideals considered in [23,24]. Note that the class of G-ideals contains the ideals generated by α-densities with α ≥ −1 (in particular, I d and the collection of logarithmic density zero sets), several summable ideals, and the Pólya ideal, i.e., Section 2].…”

Achievement sets and sum ranges with ideal supports

“…Indeed, note that | p n=1 (x n − x σ(n) ) − x| ≥ | p+1 n=1 (x n − x σ(n) ) − x| if in steps p and p + 1 the same condition ((1) or (2)) is fulfilled. On the other hand if between steps p and p + 1 the condition changes (from (1) to (2)…”