A More Sums Than Differences (MSTD) set is a set of integers A ⊂ {0, . . . , n − 1} whose sumset A + A is larger than its difference set A − A. While it is known that as n → ∞ a positive percentage of subsets of {0, . . . , n − 1} are MSTD sets, the methods to prove this are probabilistic and do not yield nice, explicit constructions. Recently Miller, Orosz and Scheinerman [MOS] gave explicit constructions of a large family of MSTD sets; though their density is less than a positive percentage, their family's density among subsets of {0, . . . , n − 1} is at least C/n 4 for some C > 0, significantly larger than the previous constructions, which were on the order of 1/2 n/2 . We generalize their method and explicitly construct a large family of sets A with |A + A + A + A| > |(A + A) − (A + A)|. The additional sums and differences allow us greater freedom than in [MOS], and we find that for any ǫ > 0 the density of such sets is at least C/n ǫ . In the course of constructing such sets we find that for any integer k there is an A such that |A+A+A+A|−|A+A−A−A| = k, and show that the minimum span of such a set is 30.
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