Lévy group and density measures

Sleziak, Martin; Ziman, Miloš

doi:10.1016/j.jnt.2008.05.004

Cited by 19 publications

(17 citation statements)

References 10 publications

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“…Similar goals have been pursued by several authors in the past, though to the best of our knowledge early work on the subject has been mostly focused on the investigation of densities raising as a limit (in a broad sense) of a sequence or a net of measures, see, e.g., R. C. Buck [4], R. Alexander [1], D. Maharam [26], T. Šalát and R. Tijdeman [37], A. H. Mekler [28], A. Fuchs and R. Giuliano Antonini [9], A. Blass, R. Frankiewicz, G. Plebanek, and C. Ryll-Nardzewski [2], M. Sleziak and M. Ziman [40], and M. Di Nasso [5]. On the other hand, M. Di Nasso and R. Jin [6] have very recently proposed a notion of "abstract upper density", which, though much coarser than our notion of upper density, encompasses a significantly larger number of upper (and lower) densities commonly considered in number theory.…”

Section: Introductionmentioning

confidence: 99%

On the notions of upper and lower density

Leonetti¹,

Tringali²

2019

Proceedings of the Edinburgh Mathematical Society

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Let P(N) be the power set of N. We say that a function µ ⋆ : P(N) → R is an upper density if, for all X, Y ⊆ N and h, k ∈ N + , the following hold: (f1) µ ⋆ (N) = 1; (f2)We show that the upper asymptotic, upper logarithmic, upper Banach, upper Buck, upper Pólya, and upper analytic densities, together with all upper α-densities (with α a real parameter ≥ −1), are upper densities in the sense of our definition. Moreover, we establish the mutual independence of axioms (f1)-(f5), and we investigate various properties of upper densities (and related functions) under the assumption that (f2) is replaced by the weaker condition that µ ⋆ (X) ≤ 1 for every X ⊆ N.Overall, this allows us to extend and generalize results so far independently derived for some of the classical upper densities mentioned above, thus introducing a certain amount of unification into the theory.

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Section: Introductionmentioning

confidence: 99%

On the notions of upper and lower density

Leonetti¹,

Tringali²

2019

Proceedings of the Edinburgh Mathematical Society

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“…This is a finer version of Lemma 5 of (Lauwers, 1988), which is a generalization of Lemma 3 of (Blumlinger, 1996) and Lemma 2.4 of Obata (1988). Sleziak and Ziman gave a proof of this in (Sleziak and Ziman, 2007). …”

Section: Countably Many Votersmentioning

confidence: 86%

May’s theorem in an infinite setting

Surekha

Rao

2010

Journal of Mathematical Economics

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“…Note that σ already appeared in the literature: indeed, it has been shown in [24] that a permutation σ belongs to the Lévy group, i.e., σ satisfies lim n [24,Theorem 2.3] for a related result.…”

Section: 2mentioning

confidence: 99%

A characterization of $(\mathcal{I}, \mathcal{J})$-regular matrices

Connor¹,

Leonetti²

2020

Preprint

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Let I, J be two ideals on N which contain the family Fin of finite sets. We provide necessary and sufficient conditions on the entries of an infinite real matrix A = (a n,k ) which maps I-convergent bounded sequences into J -convergent bounded sequences and preserves the corresponding ideal limits. The well-known characterization of regular matrices due to Silverman-Toeplitz corresponds to the case I = J = Fin.Lastly, we provide some applications to permutation and diagonal matrices, which extend several known results in the literature.Theorem 1.1. A matrix A is regular if and only if :The aim of this work is to generalize Theorem 1.1 in the context of ideal convergence.Recall that an ideal I ⊆ P(N) is a family of subsets of N closed under taking finite unions and subsets. Unless otherwise stated, it is also assumed that I is admissible, i.e., it contains the family of finite sets Fin and I = P(N). Let I ⋆ = {A ⊆ N : A c ∈ I} be its dual filter. An important example of ideal is the family of asymptotic density zero sets, that is,Accordingly, Z-convergence is usually termed statistical convergence. We let c(I) be the vector space I-convergent sequences and c 0 (I) be its subspace of sequences with I-limit 0. Structural properties of the set of bounded I-convergent sequences c(I) ∩ ℓ ∞ and its subspace c 0 (I) ∩ ℓ ∞ have been recently studied in the literature, sometimes providing answers to longstanding questions, see e.g. [5,14,18,22].

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Lévy group and density measures

Cited by 19 publications

References 10 publications

On the notions of upper and lower density

On the notions of upper and lower density

May’s theorem in an infinite setting

A characterization of $(\mathcal{I}, \mathcal{J})$-regular matrices

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