On structure of the family of regularly distributed sets with respect to the union

Abstract

Let 0 ≤ 𝑞 ≤ 1 and N denotes the set of all positive integers. In this paper we will be interested in the family 𝒰(𝑥 𝑞 ) of all regularly distributed set 𝑋 ⊂ N whose ratio block sequence is asymptotically distributed with distribution function 𝑔(𝑥) = 𝑥 𝑞 ; 𝑥 ∈ (0, 1], and we will study the structure of this family with respect to the union.