Restricted Sumsets in Finite Vector Spaces: The Case p=3

We determine the sharp lower bound for the cardinality of the restricted sumset A+' B = {a + b | a ∈ A, b ∈ B, a \neq b}, where A, B run over all subsets of size r = s =1+3h in a vector space over F3. This solves a conjecture stated in an earlier paper of ours on sumsets and restricted sumsets in finite vector spaces. The analogous problem for an arbitrary prime p remains open. However, we do prove some partial results concerning more generally special pairs of the form r = s =1+ aph. We also provide alternate proofs for the formulas satisfied by our general lower bounds βp(r, s) and γp(r, s) for the cardinality of the ordinary sum and restricted sum of sets of size r, s in a vector space over Fp