Finding and Counting MSTD Sets

Abstract

We review the basic theory of More Sums Than Differences (MSTD) sets, specifically their existence, simple constructions of infinite families, the proof that a positive percentage of sets under the uniform binomial model are MSTD but not if the probability that each element is chosen tends to zero, and 'explicit' constructions of large families of MSTD sets. We conclude with some new constructions and results of generalized MSTD sets, including among other items results on a positive percentage of sets having a given linear combination greater than another linear combination, and a proof that a positive percentage of sets are k-generational sum-dominant (meaning A, A + A, . . . , kA = A + · · · + A are each sum-dominant).

Explicit Constructions of Large Families of Generalized More Sums Than Differences Sets

“…They showed the percentage is at least 2 · 10 −7 , which was improved by Zhao [Zh2] to at least 4.28 · 10 −4 (Monte Carlo simulations suggest that approximately 4.5 · 10 −4 percent are sum-dominant). See [ILMZ1] for a survey of the field, where these and other results (such as those in [HM1,HM2], which deal with varying the probability measure on {0, . .…”

“…For definiteness in this paper we mostly study sets with |A + A + A + A| > |A + A − A − A|, and we give an example where this holds. For general comparisons, Iyer, Lazarev, Miller and Zhang recently proved existence and positive percentage (see [ILMZ1,ILMZ2] for the construction). Our main result is the following.…”