Difference bases in dihedral groups

A subset $B$ of a group $G$ is called a {em‎ ‎difference basis} of $G$ if each element $gin G$ can be written as the‎ ‎difference $g=ab^{-1}$ of some elements $a,bin B$‎. ‎The smallest‎ ‎cardinality $|B|$ of a difference basis $Bsubset G$ is called the {em‎ ‎difference size} of $G$ and is denoted by $Delta[G]$‎. ‎The fraction ‎‎‎$eth[G]:=Delta[G]/{sqrt{|G|}}$ is called the {em difference characteristic} of $G$‎. ‎We prove that for every $nin N$ the dihedral group‎ ‎$D_{2n}$ of order $2n$ has the difference characteristic‎ ‎$sqrt{2}leeth[D_{2n}]leqfrac{48}{sqrt{586}}approx1.983$‎. ‎Moreover‎, ‎if $nge 2cdot 10^{15}$‎, ‎then $eth[D_{2n}]