A structure theorem for sets of small popular doubling

TL;DR

This paper establishes a structure theorem for sets with small popular doubling, showing they are close to arithmetic progressions, and applies this to improve probability estimates for large gaps in sumsets.

Contribution

It proves a new structure theorem for sets with small popular doubling and refines probability bounds for large gaps in sumsets.

Findings

Sets with small popular doubling are close to arithmetic progressions.

Improved estimate: probability that a random subset has a large gap in its sumset is Θ(2^{-k/2}).

Enhanced understanding of sumset structure and probabilistic behavior.

Abstract

In this paper we prove that every set $A\subset\mathbb{Z}$ satisfying the inequality $\sum_{x}\min(1_A*1_A(x),t)\le(2+\delta)t|A|$ for $t$ and $\delta$ in suitable ranges, then $A$ must be very close to an arithmetic progression. We use this result to improve the estimates of Green and Morris for the probability that a random subset $A\subset\mathbb{N}$ satisfies $|\mathbb{N}\setminus(A+A)|\ge k$; specifically we show that $\mathbb{P}(|\mathbb{N}\setminus(A+A)|\ge k)=\Theta(2^{-k/2})$.