D ec 2 01 1 GENERALIZED MORE SUMS THAN DIFFERENCES SETS

A More Sums Than Differences (MSTD, or sum-dominant) set is a finite set A ⊂ Z such that |A+A| < |A−A|. Though it was believed that the percentage of subsets of {0, . . . , n} that are sum-dominant tends to zero, in 2006 Martin and O’Bryant [MO] proved that a positive percentage are sum-dominant. We generalize their result to the many different ways of taking sums and differences of a set. We prove that |ǫ1A+ · · ·+ ǫkA| > |δ1A+ · · ·+ δkA| a positive percent of the time for all nontrivial choices of ǫj , δj ∈ {−1, 1}. Previous approaches proved the existence of infinitely many such sets given the existence of one; however, no method existed to construct such a set. We develop a new, explicit construction for one such set, and then extend to a positive percentage of sets. We extend these results further, finding sets that exhibit different behavior as more sums/differences are taken. For example, we prove that for any m, |ǫ1A+· · ·+ǫkA|−|δ1A+· · ·+δkA| = m a positive percentage of the time. We find the limiting behavior of kA = A+ · · ·+A for an arbitrary set A as k → ∞ and an upper bound of k for such behavior to settle down. Finally, we say A is k-generational sum-dominant if A, A + A, . . . , kA are all sum-dominant. Numerical searches were unable to find even a 2-generational set (heuristics indicate that the probability is at most 10, and quite likely significantly less). We prove that for any k a positive percentage of sets are k-generational, and no set can be k-generational for all k.