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 n nonzero elements that I call "forbidden" elements. Let's $A$ is a subset of this vector space when the intersection of $A+A$ and $F$ is empty. The question is this: What is a non trivial lower bound for the cardinality of $A$?

For the first step I assumed that the vector space is over the field $Z_2$ and so it has just $2^n$ elements.

Thank you.