Difference bases in dihedral groups

A subset $B$ of a group $G$ is called a {\em‎ ‎difference basis} of $G$ if each element $g\in G$ can be written as the‎ ‎difference $g=ab^{-۱}$ of some elements $a,b\in B$‎. ‎The smallest‎ ‎cardinality $|B|$ of a difference basis $B\subset G$ 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 $n\in N$ the dihedral group‎ ‎$D_{۲n}$ of order $۲n$ has the difference characteristic‎ ‎$\sqrt{۲}\le\eth[D_{۲n}]\leq\frac{۴۸}{\sqrt{۵۸۶}}\approx۱.۹۸۳$‎. ‎Moreover‎, ‎if $n\ge ۲\cdot ۱۰^{۱۵}$‎, ‎then $\eth[D_{۲n}]<\frac{۴}{\sqrt{۶}}\approx۱.۶۳۳$‎. ‎Also we calculate the difference sizes and characteristics of all dihedral groups of cardinality $\le۸۰$‎.

Keywords:

‎dihedral group‎ , ‎difference basis‎ , ‎difference characteristic