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

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.