Duality between measure and category of almost all subsequences of a given sequence

Let $$\mathcal {I}$$ I be a meager ideal on $$\mathbf {N}$$ N . We show that if x is a sequence with values in a separable metric space then the set of subsequences [resp. permutations] of x which preserve the set of $$\mathcal {I}$$ I -cluster points of x is topologically large if and only if every ordinary limit point of x is also an $$\mathcal {I}$$ I -cluster point of x. The analogue statement fails for all maximal ideals. This extends the main results in [Topology Appl. 263 (2019), 221–229]. As an application, if x is a sequence with values in a first countable compact space which is $$\mathcal {I}$$ I -convergent to $$\ell $$ ℓ , then the set of subsequences [resp. permutations] which are $$\mathcal {I}$$ I -convergent to $$\ell $$ ℓ is topologically large if and only if x is convergent to $$\ell $$ ℓ in the ordinary sense. Analogous results hold for $$\mathcal {I}$$ I -limit points, provided $$\mathcal {I}$$ I is an analytic P-ideal.

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“…is comeager, cf. also [27] for the case I = Z and [22] for a measure theoretic analogue. We will extend this result to all meager ideals.…”

Section: I-cluster Pointsmentioning

confidence: 99%

The Baire Category of Subsequences and Permutations which preserve Limit Points

Balcerzak

Leonetti

2020

Results Math

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“…is comeager, cf. also [24] for the case I = Z and [20] for a measure theoretic analogue. We will extend this result to all meager ideals.…”

Section: Resultsmentioning

confidence: 99%

The Baire Category of Subsequences and Permutations which preserve Limit Points

Balcerzak¹,

Leonetti²

2020

Preprint

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Let I be a meager ideal on N. We show that if x is a sequence with values in a separable metric space then the set of subsequences [resp. permutations] of x which preserve the set of I-cluster points of x is topologically large if and only if every ordinary limit point of x is also an I-cluster point of x. The analogue statement fails for all maximal ideals. This extends the main results in [Topology Appl. 263 (2019), 221-229]. As an application, if x is a sequence with values in a first countable compact space which is I-convergent to ℓ, then the set of subsequences [resp. permutations] which are I-convergent to ℓ is topologically large if and only if x is convergent to ℓ in the ordinary sense. Analogous results hold for I-limit points, provided I is an analytic P-ideal.

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“…It is known that the families I x defined in (1) are "small", both in the measure-theoretic sense and the categorical sense, meaning that "almost all" subseries diverge, see [3,6,13,16]. Related results in the context of filter convergence have been given in [1,2,10].…”

Section: Introductionmentioning

confidence: 99%

Convergent subseries of divergent series

Balcerzak

Leonetti

2021

Rend. Circ. Mat. Palermo, II. Ser

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Let $$\mathscr {X}$$ X be the set of positive real sequences $$x=(x_n)$$ x = ( x n ) such that the series $$\sum _n x_n$$ ∑ n x n is divergent. For each $$x \in \mathscr {X}$$ x ∈ X , let $$\mathcal {I}_x$$ I x be the collection of all $$A\subseteq \mathbf {N}$$ A ⊆ N such that the subseries $$\sum _{n \in A}x_n$$ ∑ n ∈ A x n is convergent. Moreover, let $$\mathscr {A}$$ A be the set of sequences $$x \in \mathscr {X}$$ x ∈ X such that $$\lim _n x_n=0$$ lim n x n = 0 and $$\mathcal {I}_x\ne \mathcal {I}_y$$ I x ≠ I y for all sequences $$y=(y_n) \in \mathscr {X}$$ y = ( y n ) ∈ X with $$\liminf _n y_{n+1}/y_n>0$$ lim inf n y n + 1 / y n > 0 . We show that $$\mathscr {A}$$ A is comeager and that contains uncountably many sequences x which generate pairwise nonisomorphic ideals $$\mathcal {I}_x$$ I x . This answers, in particular, an open question recently posed by M. Filipczak and G. Horbaczewska.

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Duality between measure and category of almost all subsequences of a given sequence

Cited by 15 publications

References 14 publications

The Baire Category of Subsequences and Permutations which preserve Limit Points

The Baire Category of Subsequences and Permutations which preserve Limit Points

The Baire Category of Subsequences and Permutations which preserve Limit Points

Convergent subseries of divergent series

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