Sets Characterized by Missing Sums and Differences

TL;DR

This paper investigates the probability and structure of MSTD sets, showing it converges to a specific limit and providing algorithms and insights into their formation.

Contribution

It improves the known lower bound for the probability of MSTD sets and introduces a deterministic algorithm to compute this probability precisely.

Findings

Probability of MSTD sets approaches approximately 4.5 x 10^{-4}.

Middle elements in MSTD sets are nearly always included as n grows large.

Fringe elements significantly influence the set's sum-difference properties.

Abstract

A more sums than differences (MSTD) set is a finite subset S of the integers such |S+S| > |S-S|. We show that the probability that a uniform random subset of {0, 1, ..., n} is an MSTD set approaches some limit rho > 4.28 x 10^{-4}. This improves the previous result of Martin and O'Bryant that there is a lower limit of at least 2 x 10^{-7}. Monte Carlo experiments suggest that rho \approx 4.5 \x 10^{-4}. We present a deterministic algorithm that can compute rho up to arbitrary precision. We also describe the structure of a random MSTD subset S of {0, 1, ..., n}. We formalize the intuition that fringe elements are most significant, while middle elements are nearly unrestricted. For instance, the probability that any ``middle'' element is in S approaches 1/2 as n -> infinity, confirming a conjecture of Miller, Orosz, and Scheinerman. In general, our results work for any specification on the number of missing sums and the number of missing differences of S, with MSTD sets being a special case.