Generalized More Sums Than Differences sets

TextA More Sums Than Differences (or sum-dominant) set is a finite set A⊂ZA⊂Z with |A+A|<|A−A||A+A|<|A−A|. Though it was believed that the percentage of subsets of {0,…,n}{0,…,n} that are sum-dominant tends to zero, Martin and OʼBryant proved a positive percentage is sum-dominant. We generalize their result to other sums and differences. We prove that |ϵ1A+⋯+ϵkA|>|δ1A+⋯+δkA||ϵ1A+⋯+ϵkA|>|δ1A+⋯+δkA| a positive percent of the time for all nontrivial choices of ϵj,δj∈{−1,1}ϵj,δj∈{−1,1}, and give explicit constructions. We construct sets exhibiting different behavior as more sums/differences are taken. We prove that for any m  , |ϵ1A+⋯+ϵkA|−|δ1A+⋯+δkA|=m|ϵ1A+⋯+ϵkA|−|δ1A+⋯+δkA|=m a positive percentage of the time. We find the limiting behavior of kA=A+⋯+AkA=A+⋯+A for an arbitrary set A   as k→∞k→∞ and an upper bound on k for such behavior to settle down. Finally, we say A is k  -generational sum-dominant if A,A+A,…,kAA,A+A,…,kA are all sum-dominant. We prove that for any k a positive percentage of sets is k-generational, and no set is k-generational for all k.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=ERSMlrEAijY;list=UUfJicAn0WSCOS0IZWMy7HsA;index=1;feature=plcp.