Asymptotic Nonexistence of Difference Sets in Dihedral Groups

Almost all known results on difference sets need severe restrictions on the parameters. The main purpose of this paper is to provide an asymptotic nonexistence result free from any assumptions on the parameters. The only assumption we make is that the underlying group is dihedral. Difference sets originally mainly were studied in cyclic groups where they exist in abundance. For example, for any prime power n; there is a difference set in the cyclic group of order n þ nþ 1; namely, the so-called Singer difference set, see [2]. It is a very interesting phenomenon that the situation changes completely if one switches from cyclic groups to dihedral groups. No nontrivial difference set in any dihedral group has been found yet.