I have a set $S$ of positive integers, and would like to prove that all large enough $n$ are of the form $s+t$ with $s,t\in S.$ (In other words, $\mathbb{N} \setminus (S+S)$ is finite, where $+$ is the sumset.) What techniques can I use to solve this sort of problem?
My particular example seems simple because not only is $S$ is dense $$ \liminf_n\frac{\#\left(S\cap\{1,2,\ldots,n\}\right)}{n}>0 $$ but it has another largeness property (I don't know a name for this): for every residue class $a\pmod b$, $$ \liminf_n\frac{\#\left(S\cap\{a,a+b,a+2b,\ldots,a+nb\}\right)}{a+nb}>0. $$ Is this property sufficient? Given a set $S$, can a reasonable bound be given for the greatest integer not in $S+S$?
If the proof is easy, I'd be happy for just hints (it's nice to prove things for myself!). Otherwise I'd be happy with a reference.
The two conditions you list are insufficient. If $\alpha$ is irrational and $\{x\}$ denotes the fractional part of the real number $x$, then the set $A:=\{n \in \mathbb N: \{n\alpha\} \in (0,1/4)\}$ has $\lim_{n\to \infty} \frac{|A\cap \{a+b,\dots, a+bn\}|}{n}= 1/4$ for every $a, b\in \mathbb N$, while $A+A$ is not cofinite. In fact, $A+A$ is contained in $\{n\in \mathbb N: \{n\alpha\} \in (0,1/2)\}$, which has asymptotic density $1/2$.