Let $S$ be a set of nonnegative integers such that the sumset $S+S$ is the nonnegative integers. If it can, what is the fastest growing function $f$ such that the number of elements of $S$ less than $n$ is proportional to $n/f(n)?$
$S$ can have zero natural density, with an example being the numbers that are sums of two nonnegative integer squares, but it has $f = \sqrt{\log {n}},$ so it looks suboptimal to me.