On the Local Distribution of Elements of Subsets of the Set of Positive Integers

For an infinite set $$\mathscr A\subset\mathbb N$$ , let the function $$\rho(x,y;\mathscr A)$$ , where $$x\in\mathbb N$$ and $$y+1\in\mathbb N$$ , be defined as the length of a maximal interval $$(\alpha,\beta)\subset(y,x+y)$$ disjoint from $$\mathscr A$$ . We study the function $$\rho_*(x;\mathscr A)=\inf_{y+1\in\mathbb N}\rho(x,y;\mathscr A).$$ In particular, we determine the order of this function for the set $$\mathscr A$$ of square-free numbers.