This is problem 1.1.6 from the book "Additive Combinatorics" by Tao and Vu.
Suppose A is a subset of an additive group Z. We need to show that there exists a d-element subset of Z, denoted B = { v1, v2, ... vd } with d = O( log |Z|/|A| ) such that A + FS(B) is of size at least |Z|/2. ( FS(B) is the subset sums of B)
So I tried taking a random d-element subset of Z and a specific z in Z so that I can bound the probability that z is contained in A + FS(B) and then apply the first moment to get a bound on the average size of the set. But I am unable to get a bound. Please help if possible. (This is not part of any homework.)