Sets with more differences than sums

TL;DR

This paper proves that for a random set of integers with density zero, the number of differences exceeds the number of sums almost always, confirming a conjecture by Martin and O'Bryant.

Contribution

It establishes that sparse random sets of integers typically have more differences than sums, confirming a longstanding conjecture.

Findings

Random sets with density zero have more differences than sums almost surely.

The conjecture by Martin and O'Bryant is proven.

The result applies to a broad class of sparse sets.

Abstract

We show that a random set of integers with density 0 has almost always more differences than sums. This proves a conjecture by Martin and O'Bryant.