Explicit constructions of infinite families of MSTD sets

TL;DR

This paper presents explicit constructions of infinite families of MSTD sets, demonstrating they are more common than previously thought and providing a method to compare linear forms of sets.

Contribution

The authors introduce a new explicit construction method for infinite MSTD sets and establish their higher density compared to earlier known families.

Findings

Existence of a constant C such that at least C/r^4 of subsets are MSTD

Constructed families are significantly denser than previous ones

Generalized method to compare linear forms of sets

Abstract

We explicitly construct infinite families of MSTD (more sums than differences) sets. There are enough of these sets to prove that there exists a constant C such that at least C / r^4 of the 2^r subsets of {1,...,r} are MSTD sets; thus our family is significantly denser than previous constructions (whose densities are at most f(r)/2^{r/2} for some polynomial f(r)). We conclude by generalizing our method to compare linear forms epsilon_1 A + ... + epsilon_n A with epsilon_i in {-1,1}.