Fringe Pairs in Generalized Mstd Sets
نویسندگان
چکیده
A More Sums Than Differences (MSTD) set is a set A for which |A+A| > |A−A|. Martin and O’Bryant proved that the proportion of MSTD sets in {0, 1, . . . , n} is bounded below by a positive number as n goes to infinity. Iyer, Lazarev, Miller and Zhang introduced the notion of a generalized MSTD set, a set A for which |sA− dA| > |σA − δA| for a prescribed s + d = σ + δ. We offer efficient constructions of k-generational MSTD sets, sets A where A,A + A, . . . , kA are all MSTD. We also offer an alternative proof that the proportion of sets A for which |sA−dA|−|σA−δA|= x is positive, for any x ∈ Z. We prove that for any ǫ > 0, Pr(1−ǫ < log |sA−dA|/ log |σA−δA| < 1+ǫ) goes to 1 as the size of A goes to infinity and we give a set A which has the current highest value of log |A+A|/ log |A−A|. We also study decompositions of intervals {0, 1, . . . , n} into MSTD sets and prove that a positive proportion of decompositions into two sets have the property that both sets are MSTD.
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