A structure theorem for sets of small popular doubling
Volume 171 / 2015
Acta Arithmetica 171 (2015), 221-239
MSC: Primary 11P70.
DOI: 10.4064/aa171-3-2
Abstract
We prove that every set $A\subset\mathbb{Z}$ satisfying $\sum_{x}\min(1_A*1_A(x),t)\le (2+\delta)t|A|$ for $t$ and $\delta$ in suitable ranges 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 $\mathbb{P}(|\mathbb{N}\setminus(A+A)|\ge k)=\varTheta(2^{-k/2})$.
