A structure theorem for sets of small popular doubling

Przemysław Mazur¹

¹ Mathematical Institute Radcliffe Observatory Quarter Woodstock Road, Oxford OX2 6GG, United Kingdom

Acta Arithmetica, Tome 171 (2015) no. 3, pp. 221-239.

Résumé

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 $\mathbb{P}(|\mathbb{N}\setminus(A+A)|\ge k)=\varTheta(2^{-k/2})$.

Détail

DOI : 10.4064/aa171-3-2

Keywords: prove every set subset mathbb satisfying sum min a* delta delta suitable ranges close arithmetic progression result improve estimates green morris probability random subset subset mathbb satisfies mathbb setminus specifically mathbb mathbb setminus vartheta k