The difference basis and bi-basis of ZmZm

TextFor each positive integer m  , let σA(n)=♯{(a,a′)∈A2:a+a′=n}, δA(n)=♯{(a,a′)∈A2:a−a′=n}, where n∈Zmn∈Zm and A   is a subset of ZmZm. Recently Chen proved that for each positive integer m  , there exists a set A⊆ZmA⊆Zm such that A+A=ZmA+A=Zm and σA(n)⩽288σA(n)⩽288 for any n∈Zmn∈Zm. In this paper, the following results are proved: (i) for each positive integer m  , there exists a set A⊆ZmA⊆Zm such that Zm=A−AZm=A−A and δA(n)⩽7δA(n)⩽7 for all n∈Zmn∈Zm with at most 3 exceptions; (ii) for each positive integer m  , there exists a set A′⊆ZmA′⊆Zm with A′+A′=ZmA′+A′=Zm and A′−A′=ZmA′−A′=Zm such that σA′(n)⩽26σA′(n)⩽26 and δA′(n)⩽24δA′(n)⩽24 for all n∈Zmn∈Zm with at most 3 exceptions.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=GfZ07Fg5qXE.