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 $A$?
