I do not know whether this is the right place for posting this problem. But for several months I have no solution to this problem.

Let $A$ be Lebesgue measurable subset of $[0,\infty)$ such that Lebesgue measure of $A$ is positive i.e. $0<\lambda(A)<\infty$. Let $S$ be the set defined as follows: $$S:=\{t\in [0,\infty):nt\in A\text{ for infinitely many }n\in\Bbb N\}$$

What can we conclude about the measure of $S$?

I can guess that $\lambda (S)=0$ for when $A$ is an open set but can't prove it. More particular case, when $A$ is open with finitely many components then I can conclude that $\lambda(S)=0$