Characteristics of distributions of sets and their $(R)$- and $(N)$-denseness

TL;DR

This paper explores the properties and relationships of regularly distributed sets within natural numbers, establishing equivalences among various notions like regular variation, denseness, and distribution functions.

Contribution

It introduces a unified framework linking regular distribution, denseness, and related concepts in the context of sets of natural numbers.

Findings

Regular distributed sets are equivalent to regular sequences and regular variation.

The paper establishes relationships between (N)-denseness, direction sets, and dispersion.

Distribution functions of the form g(x) = x^q characterize the asymptotic distribution of sets.

Abstract

Let $0\leq q\leq1$ and $\mathbb{N}$ denotes the set of all positive integers. In this paper we will deal with it too the family $\mathcal{U}(x^q)$ of all regularly distributed set $X \subset \mathbb{N}$ whose ratio block sequence is asymptotically distributed with distribution function $g(x) = x^q;\ x \in(0,1]$, and we will show that the regular distributed set, regular sequences, regular variation at infinity are equivalent notations. In this paper also we discuss the relation ship between notations as (N)-denseness, directions sets, generalized ratio sets, dispersion of sequence and exponent of convergence.