Thinking about the four square theorem and related questions, I found myself wondering: What is the minimal density of a set $A \subset \{0, 1, 2, ... \}$ such that $A + A = \mathbb{N}$?

What I know:

1. If A has less than quadratic density, then $A + A$ is not $\mathbb{N}$ by a simple counting argument.
2. There are quadratic density sets $A$ such that $A + A + A$ is $\mathbb{N}$, such as the triangular numbers.
3. For any positive constant $\varepsilon > 0$ there is a set of density $\varepsilon$ satisfying $A + A = \mathbb{N}$: Let $k = \lceil 1/\varepsilon \rceil$, and set $A = [k-1] \cup \{ kn : n \in \mathbb{N} \}$.