On structure of the family of regularly distributed sets

Let $$0\le q\le 1$$ and $${\mathbb {N}}$$ denote the set of all positive integers. In this paper (which is a continuation of [17]) we will be interested in the family $${\mathcal {U}}(x^q)$$ of all regularly distributed sets $$X \subset {\mathbb {N}}$$ whose ratio block sequence of type (1.1) is asymptotically distributed with distribution function $$g(x) = x^q;\ x \in [0,1]$$ . We will study the structure of these families with respect to the union, intersection and difference of the sets $$X\in {\mathcal {U}}(x^p)$$ , $$Y\in {\mathcal {U}}(x^q)$$ , where $$0\le p, q\le 1.$$