For $A\subseteq\mathbb{N}$ we define the upper density to be $$\text{ud}(A) = \lim\sup\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}$$


Is there an infinite set ${\cal S}$ of pairwise disjoint subsets of $\mathbb{N}$ such that $\text{ud}(S) > 0$ for all $S\in {\cal S}$?



Pairwise disjoint subsets of $\mathbb{N}$ with positive upper density
Yes. Take the segments of integers $B\_i=\{2^i,\dots,2^{i+1}-1\}$. Partition naturals onto disjoint infinite sets $I\_1,\dots$. Take the sets $A\_i=\cup\_{k\in I\_i} B\_k$.


