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 $\lambda(S)=0$
Define for $N\in\mathbb{N}$: $$S_N=\{t\in[0,N): nt\in A\text{ for infinitely many } n\in\mathbb{N}\}.$$ Then $nS_N\backslash S_N\subset A$ for infinitely many $n\in\mathbb{N}$ and $\lambda(nS_N\backslash S_N)\geq\lambda(S_N)$ for $n$ sufficiently large. Continuing in this way You can construct a sequence of disjoint subsets of $A$ of measure at least $\lambda(S_N)$ and thus since $\lambda(A)<\infty$ You can conclude $\lambda (S_N)=0$. It follows $$\lambda(S)=\lambda(\bigcup_{N=1}^{\infty}S_N)=0.$$