Assume that you have an $n$-dimensional vector space over a finite field (therefore the number of elements in the vector space is finite) and $F$ is a subset of this vector space which contains $m$ elements. Let's $A$ is a subset of this vector space such that the intersection of $A+A$ and $F$ is empty.

The question is: What is a non trivial lower bound for the cardinality of $A$?

Thank you.