Suppose $A$ is a subset of the finite field with $p$ elements. What is the best approximation of $A$ by a sumset $B+C$ in the sense that $A\Delta (B+C)$ is minimal? Of course if $B=Ax$ and $C=\{x\}$ then we have equality so I would ask that both sets be nonsingleton. If $A+A$ is small then there are standard covering lemmas which give an answer but this is a fairly strong assumption about $A$. If I recall correctly there are results that say $A$ is a sumset if it is almost the whole field. What about smaller $A$?
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$\begingroup$ For small B and C I can imagine perturbing B+C in many places to get about half of it away from B+C, and likely a quarter of it from any other sumset, but I do not have a construction. It might be prudent to count the fraction of p choose 12 subsets of size 12 that are nontrivial sumsets. Gerhard "Likes Working With Small Examples" Paseman, 2013.06.10 $\endgroup$– Gerhard PasemanCommented Jun 10, 2013 at 17:35
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