Sequences of integers with missing quotients and dense points without neighbors

Let A be a pre-defined set of rational numbers. We say that a set of natural numbers S is an A -quotient-free set if no ratio of two elements in S belongs to A . We find the maximal asymptotic density and the maximal upper asymptotic density of A -quotient-free sets when A belongs to a particular class. known that in the case A = { p , q } , where p , q are coprime integers greater than 1, the latter problem is reduced to the evaluation of the largest number of non-adjacent lattice points in a triangle whose legs lie on the coordinate axes. We prove that this number is achieved by choosing points of the same color in the checkerboard coloring.