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

TL;DR

This paper provides explicit constructions of large families of sets with more sums than differences, extending previous work to more complex sumsets and difference sets, with implications for combinatorial number theory.

Contribution

It generalizes existing explicit constructions of MSTD sets to more complex sumsets, achieving higher density and greater flexibility in set properties.

Findings

Constructed sets with |A+A+A+A| > |(A+A)-(A+A)|

Density of such sets is at least C / n^epsilon for any epsilon>0

Found sets where the difference |A+A+A+A| - |A+A-A-A| equals any integer k, with minimum span 30.

Abstract

A More Sums Than Differences (MSTD) set is a set of integers A contained in {0, ..., n-1} whose sumset A+A is larger than its difference set A-A. While it is known that as n tends to infinity a positive percentage of subsets of {0, ..., n-1} are MSTD sets, the methods to prove this are probabilistic and do not yield nice, explicit constructions. Recently Miller, Orosz and Scheinerman gave explicit constructions of a large family of MSTD sets; though their density is less than a positive percentage, their family's density among subsets of {0, ..., n-1} is at least C/n^4 for some C>0, significantly larger than the previous constructions, which were on the order of 1 / 2^{n/2}. We generalize their method and explicitly construct a large family of sets A with |A+A+A+A| > |(A+A)-(A+A)|. The additional sums and differences allow us greater freedom than in Miller, Orosz and Scheinerman, and we find that for any epsilon>0 the density of such sets is at least C / n^epsilon. In the course of constructing such sets we find that for any integer k there is an A such that |A+A+A+A| - |A+A-A-A| = k, and show that the minimum span of such a set is 30.