The theory of sumsets $A+B$ where $A$ and $B$ are finite subsets of an additive group $Z$ is extensively studied in additive combinatorics: finding long arithmetic progressions inside them, finding lots of subsets of this form, bounding its size above and below, and so on.


A fairly natural inverse question is the following.



> 
> Is there a function $f$ such that if $\lvert A\rvert\gg f(\lvert Z\rvert)$ then $A=B+C$ where $B$ and $C$ are both fairly large sets?
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> 
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Since results such as Szemeredi's theorem and Ramsey theory suggest that sets can have lots of structure from cardinality conditions alone, and sumsets are very structured, this seems like a plausible hope. 


The case for general finite additive groups may be too hard/trivially false, so I am (as usual in these questions) mostly interested in the cases $Z=\mathbb{Z}/N\mathbb{Z}$ and $Z=\mathbb{F}\_p^n$.


I suspect that this sort of result is already known, or follows easily from another well known theorem, and would appreciate any reference and/or proof.



How large can a non-sumset be?
As far as I know, the only result of this sort is due to Alon, found [here](http://www.tau.ac.il/~nogaa/PDFS/sumset.pdf).


