Constructing MSTD Sets Using Bidirectional Ballot Sequences

TL;DR

This paper introduces a new method for constructing dense families of MSTD sets, significantly increasing the number of such sets compared to previous methods, with implications for additive number theory.

Contribution

It presents a novel construction of MSTD sets using bidirectional ballot sequences, improving the known lower bound on the number of MSTD sets.

Findings

Constructs Theta(2^n/n) MSTD sets, surpassing previous Omega(2^n/n^4) bounds.

Provides a new combinatorial approach using bidirectional ballot sequences.

Enhances understanding of the density and distribution of MSTD sets.

Abstract

A more sums than differences (MSTD) set is a finite subset S of the integers such that |S+S| > |S-S|. We construct a new dense family of MSTD subsets of {0, 1, 2, ..., n-1}. Our construction gives Theta(2^n/n) MSTD sets, improving the previous best construction with Omega(2^n/n^4) MSTD sets by Miller, Orosz, and Scheinerman.