Influence of different kind of thin sets in the theory of convergence

TL;DR

This paper introduces very thin sets as a new class of subsets of natural numbers, proposing a modified convergence theory that overcomes limitations of statistical convergence by emphasizing these sets.

Contribution

It develops the theory of very thin sets and related subclasses, offering a new perspective on convergence in analysis.

Findings

Very thin sets can replace thin or finite sets in convergence theory.

The new framework removes some drawbacks of statistical convergence.

Concepts of super thin, very very thin, and super super thin sets are introduced.

Abstract

A class of subsets designated as very thin subsets of natural numbers has been studied and seen that theory of convergence may be rediscovered if very thin sets are given to play main role instead of thin or finite sets which removes some drawback of statistical convergence. While developing the theory of very thin sets, concepts of super thin, very very thin and super super thin sets are evolved spontaneously.