  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.
