2015/09/05 by Asada, Megumi, Manski, Sarah, Miller, Steven J. +1
#11K99 (secondary) #11P99 (primary) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1509.01657
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 ε>0, Pr(1-ε