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

Let 0 ≤ q ≤ 1 and N denotes the set of all positive integers. In this paper we will deal with it too the family U ( x q ) of all regularly distributed set X = { x 1 < x 2 < ⋯ < x n < ⋯ } ⊂ N whose ratio block sequence 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 ∈ ( 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.