Sumsets in dihedral groups

Let DnDn be the dihedral group of order 2n2n. For all integers r,sr,s such that 1≤r,s≤2n1≤r,s≤2n, we give an explicit upper bound for the minimal size μDn(r,s)=min|A⋅B|μDn(r,s)=min|A⋅B| of sumsets (product sets) A⋅BA⋅B, where AA and BB range over all subsets of DnDn of cardinality rr and ss respectively. It is shown by construction that μDn(r,s)μDn(r,s) is bounded above by the known value of μG(r,s)μG(r,s), where GG is any abelian group of order 2n2n. We conjecture that this upper bound is sharp, and prove that it really is if nn is a prime power.