We say that a set $A\subseteq \mathbb{N}$ has *lower density 0* if $$\text{lim inf}\_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 0.$$


Given $A,B\subseteq \mathbb{N}$ we set $A+B = \{a+b: a\in A, b\in B\}$. 


Are there $A, B\subseteq \mathbb{N}$ with lower density 0, but $A+B$ does not have lower density 0?



Sums of sets of lower density 0
It is relatively easy to prove that the set of perfect squares has asymptotic density equal to $0$. Then either the set $Q\_2 := \{x^2+y^2: x,y \in \mathbf N\}$ has positive lower asymptotic density, and we're done, or the lower density of $Q\_2$ is zero, and then we just consider that $\mathbf N = Q\_2 + Q\_2$ (by Lagrange's four-squares theorem).


By the way, it follows, e.g., from E. Landau, *Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindeszahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate*, Arch. Math. Phys. **13** (1908), 305-312 that the asymptotic density of $Q\_2$ is actually zero, but this is more than what you need to answer the question in the OP.


