Fringe pairs in generalized MSTD sets

Abstract

A More Sums Than Differences (MSTD) set is a set [Formula: see text] for which [Formula: see text]. Martin and O’Bryant proved that the proportion of MSTD sets in [Formula: see text] is bounded below by a positive number as [Formula: see text] goes to infinity. Iyer, Lazarev, Miller and Zhang introduced the notion of a generalized MSTD set, a set [Formula: see text] for which [Formula: see text] for a prescribed [Formula: see text]. We offer efficient constructions of [Formula: see text]-generational MSTD sets, sets [Formula: see text] where [Formula: see text] are all MSTD. We also offer an alternative proof that the proportion of sets [Formula: see text] for which [Formula: see text] is positive, for any [Formula: see text]. We prove that for any [Formula: see text], [Formula: see text] goes to [Formula: see text] as the size of [Formula: see text] goes to infinity and we give a set [Formula: see text] which has the current highest value of [Formula: see text]. We also study decompositions of intervals [Formula: see text] into MSTD sets and prove that a positive proportion of decompositions into two sets have the property that both sets are MSTD.