Some results on uniform statistical cluster points

Abstract

In this paper, we present some results linking the uniform statistical limit superior and inferior, almost convergence and uniform statistical convergence of a sequence. We also study the relationship between the set of uniform statistical cluster points of a given sequence and its subsequences. The results concerning uniform statistical convergence and uniform statistical cluster points presented here are also closely related to earlier results regarding statistical convergence and statistical cluster points of a sequence.

Some new insights into ideal convergence and subsequences

“…We remark that the last theorem provides a generalization of the author's earlier result regarding uniform statistical cluster points in [23], and this further indicates that both cases of measure 0 and 1 can occur (see [23]).…”

“…It is well known that every x ∈ (0, 1] has a unique binary expansion x = ∑ ∞ n=1 2 −n d n (x) such that d n (x) = 1 for infinitely many positive integers n, and for every x ∈ (0, 1] and any sequence s = (s n ) we can generate a subsequence (sx) of s in such a way that: if d n (x) = 1, then (sx) n = s n . In the existing literature the relationships between a given sequence and its subsequences have been studied in two directions: the first direction is changing the concept of convergence by statistical convergence, A-statistical convergence, uniform statistical convergence, ideal convergence, and the other direction is using measure or category to study the measure and topological largeness of the sets of subsequences (see [2,4,15,[18][19][20][21][22][23][24]). There are still gaps to examine in this area.…”

“…, then I d -convergence coincides with statistical convergence where d(A) denotes the natural density of A [10], and if I = I u = {A ⊂ N : u(A) = 0}, then I u -convergence reduces to uniform statistical convergence where u(A) denotes the uniform density of A [22,23]. Here, the term meager will refer to sets of first Baire category, while the term comeager will refer to sets whose complement is of first category.…”

Category theoretical view of I-cluster and I-limit points of subsequences

“…Additionally we have the following result, reminiscent of some theorems concerning statistical and uniform statistical convergence (see [15,16,20]). for n ∈ ∪ i ν=1 I 2 µ−1 (2ν+1) and any fixed µ = 1, 2, .…”