In Tao and Vu's Additive Combinatorics, it is mentioned that "if we replace the additive set $A$ by a larger set, such as $A+B$, $A+A+A$, or $2A - 2A$, then one can locate significantly larger progressions inside these sets..." Thus it would see quite interesting to investigate under what circumstances a given set is of the forms above.

On one note, there has been some work done to investigate what $B$ could be if $A, A+B$ are known, namely the complementary base problem. In particular, Vu proved in 2002 that if $\mathcal{P}$ is the set of primes, then there exists a set $B$ such that $|B \cap [1,n]| = O(\log n)$ and the set {$p + b : p \in \mathcal{P}, b \in B$} $= \mathbb{N} \setminus C$, where $C$ is a finite set.

My question is, if we know about a set $S$, are there any methods to detect if it contains or is equal to a set of the form $A+B, A+A+A, 2A-2A$ etc.?