Characteristics of Distributions of Sets and Their (R)- and (N)-Denseness

Abstract

AbstractLet $$0\le q\le 1$$ 0 ≤ q ≤ 1 and $$\mathbb {N}$$ N denotes the set of all positive integers. In this paper we will deal with it too the family $${\mathcal {U}}(x^q)$$ U ( x q ) of all regularly distributed set $$X=\{x_1<x_2<\cdots<x_n<\cdots \} \subset \mathbb {N}$$ X = { x 1 < x 2 < ⋯ < x n < ⋯ } ⊂ N whose ratio block sequence $$\begin{aligned} \frac{x_1}{x_1}, \frac{x_1}{x_2}, \frac{x_2}{x_2}, \frac{x_1}{x_3}, \frac{x_2}{x_3}, \frac{x_3}{x_3}, \dots , \frac{x_1}{x_n}, \frac{x_2}{x_n}, \dots , \frac{x_n}{x_n}, \dots \end{aligned}$$ x 1 x 1 , x 1 x 2 , x 2 x 2 , x 1 x 3 , x 2 x 3 , x 3 x 3 , ⋯ , x 1 x n , x 2 x n , ⋯ , x n x n , ⋯ is asymptotically distributed with distribution function $$g(x) = x^q;\ x \in (0,1]$$ g ( x ) = x q ; x ∈ ( 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 relationship between notations as (N)-denseness, directions sets, generalized ratio sets, dispersion and exponent of convergence.