Suppose we are given $A \subseteq \mathbb{N}$ with $\lim\sup\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} > 0$. For $k\in \mathbb{N}, k\geq 2$ we set $$M\_A(k) = \{a\in A: ka \in A\}.$$
Does there exist $k\in \mathbb{N}, k\geq 2$ such that $M\_A(k)$ is infinite?



Multiples in sets of positive upper density
Not necessarily: you can in fact have $M\_k(A)=\varnothing$ for all integer $k\ge 2$. This was shown by Besicovitch ("[On the density of certain sequences of integers](https://link.springer.com/content/pdf/10.1007%2FBF01448032.pdf)", *Math. Ann.* **110** (1935), no. 1, 336–341) who has constructed a set (of positive integers) of positive upper density such that none of the elements of the set is divisible by any other element.


