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.It is 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.