Invariance of Ideal Limit Points

Let $\mathcal{I}$ be an analytic P-ideal [respectively, a summable ideal] on the positive integers and let $(x_n)$ be a sequence taking values in a metric space $X$. First, it is shown that the set of ideal limit points of $(x_n)$ is an $F_\sigma$-set [resp., a closet set]. Let us assume that $X$ is also separable and the ideal $\mathcal{I}$ satisfies certain additional assumptions, which however includes several well-known examples, e.g., the collection of sets with zero asymptotic density, sets with zero logarithmic density, and some summable ideals. Then, it is shown that the set of ideal limit points of $(x_n)$ is equal to the set of ideal limit points of almost all its subsequences.Comment: 11 pages, no figures, to appear in Topology App

Limit points of subsequences

Let $x$ be a sequence taking values in a separable metric space and $\mathcal{I}$ be a generalized density ideal or an $F_\sigma$-ideal on the positive integers (in particular, $\mathcal{I}$ can be any Erd{\H o}s--Ulam ideal or any summable ideal). It is shown that the collection of subsequences of $x$ which preserve the set of $\mathcal{I}$-cluster points of $x$ [respectively, $\mathcal{I}$-limit points] is of second category if and only if the set of $\mathcal{I}$-cluster points of $x$ [resp., $\mathcal{I}$-limit points] coincides with the set of ordinary limit points of $x$; moreover, in this case, it is comeager. In particular, it follows that the collection of subsequences of $x$ which preserve the set of ordinary limit points of $x$ is comeager.Comment: To appear in Topology Appl. arXiv admin note: substantial text overlap with arXiv:1711.0426

Characterizations of the Ideal Core

Given an ideal $\mathcal{I}$ on $\omega$ and a sequence $x$ in a topological vector space, we let the $\mathcal{I}$-core of $x$ be the least closed convex set containing $\{x_n: n \notin I\}$ for all $I \in \mathcal{I}$. We show two characterizations of the $\mathcal{I}$-core. This implies that the $\mathcal{I}$-core of a bounded sequence in $\mathbf{R}^k$ is simply the convex hull of its $\mathcal{I}$-cluster points. As applications, we simplify and extend several results in the context of Pringsheim-convergence and $e$-convergence of double sequences.Comment: 10 pages, to appear in Journal of Mathematical Analysis and Application

Some new insights into ideal convergence and subsequences

Some results on the sets of almost convergent, statistically convergent, uniformly statistically convergent, I-convergent subsequences of (sn) have been obtained by many authors via establishing a one-to-one correspondence between the interval (0, 1] and the collection of all subsequences of a given sequence s = (sn). However, there are still some gaps in the existing literature. In this paper we plan to fill some of the gaps with new results. Some of them are easily derived from earlier results but they offer some new deeper insights. © 2022, Hacettepe University. All rights reserved

Convergence Rates of Subseries

Let $(x_n)$ be a positive real sequence decreasing to $0$ such that the series $\sum_n x_n$ is divergent and $\liminf_{n} x_{n+1}/x_n>1/2$. We show that there exists a constant $\theta \in (0,1)$ such that, for each $\ell>0$, there is a subsequence $(x_{n_k})$ for which $\sum_k x_{n_k}=\ell$ and $x_{n_k}=O(\theta^k)$.Comment: 5 pp. To appear in The American Mathematical Monthl

On the notions of upper and lower density

Let $\mathcal{P}({\bf N})$ be the power set of ${\bf N}$. We say that a function $\mu^\ast: \mathcal{P}({\bf N}) \to \bf R$ is an upper density if, for all $X,Y\subseteq{\bf N}$ and $h, k\in{\bf N}^+$, the following hold: (F1) $\mu^\ast({\bf N}) = 1$; (F2) $\mu^\ast(X) \le \mu^\ast(Y)$ if $X \subseteq Y$; (F3) $\mu^\ast(X \cup Y) \le \mu^\ast(X) + \mu^\ast(Y)$; (F4) $\mu^\ast(k\cdot X) = \frac{1}{k} \mu^\ast(X)$, where $k \cdot X:=\{kx: x \in X\}$; (F5) $\mu^\ast(X + h) = \mu^\ast(X)$. We show that the upper asymptotic, upper logarithmic, upper Banach, upper Buck, upper Polya, and upper analytic densities, together with all upper $\alpha$-densities (with $\alpha$ a real parameter $\ge -1$), 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 $\mu^\ast(X)\le 1$ for every $X\subseteq{\bf N}$. 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.Comment: 26 pp, no figs. Added a 'Note added in proof' at the end of Sect. 7 to answer Question 6. Final version to appear in Proc. Edinb. Math. Soc. (the paper is a prequel of arXiv:1510.07473

Thinnable Ideals and Invariance of Cluster Points

We define a class of so-called thinnable ideals I on the positive integers which includes several well-known examples, e.g., the collection of sets with zero asymptotic density, sets with zero logarithmic density, and several summable ideals. Given a sequence (xn) taking values in a separable metric space and a thinnable ideal I, it is shown that the set of I-cluster points of (xn) is equal to the set of I-cluster points of almost all of its subsequences, in the sense of Lebesgue measure. Lastly, we obtain a characterization of ideal convergence, which improves the main result in [15]