Finding and Counting MSTD sets

TL;DR

This paper reviews the theory of MSTD sets, discusses their existence, constructions, and properties, and introduces new results on generalized MSTD sets and their prevalence among random sets.

Contribution

It provides new constructions and results on generalized MSTD sets, including their frequency and properties in random models.

Findings

A positive percentage of sets are MSTD under the uniform binomial model.

Explicit constructions of large families of MSTD sets are provided.

A positive percentage of sets are $k$-generational sum-dominant.

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).