Sumsets in dihedral groups

Abstract

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