A structure theorem for sets of small popular doubling

Przemysław Mazur. "A structure theorem for sets of small popular doubling." Acta Arithmetica 171.3 (2015): 221-239. <http://eudml.org/doc/279419>.

@article{PrzemysławMazur2015,
abstract = {We prove that every set A ⊂ ℤ satisfying $∑_\{x\}min(1_A*1_A(x),t) ≤ (2+δ)t|A|$ for t and δ 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 ⊂ ℕ satisfies |ℕ∖(A+A)| ≥ k; specifically, we show that $ℙ(|ℕ∖(A+A)| ≥ k) = Θ(2^\{-k/2\})$.},
author = {Przemysław Mazur},
journal = {Acta Arithmetica},
keywords = {small popular doubling; structure theorem; coset; progression; regularity lemma},
language = {eng},
number = {3},
pages = {221-239},
title = {A structure theorem for sets of small popular doubling},
url = {http://eudml.org/doc/279419},
volume = {171},
year = {2015},
}

TY - JOUR
AU - Przemysław Mazur
TI - A structure theorem for sets of small popular doubling
JO - Acta Arithmetica
PY - 2015
VL - 171
IS - 3
SP - 221
EP - 239
AB - We prove that every set A ⊂ ℤ satisfying $∑_{x}min(1_A*1_A(x),t) ≤ (2+δ)t|A|$ for t and δ 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 ⊂ ℕ satisfies |ℕ∖(A+A)| ≥ k; specifically, we show that $ℙ(|ℕ∖(A+A)| ≥ k) = Θ(2^{-k/2})$.
LA - eng
KW - small popular doubling; structure theorem; coset; progression; regularity lemma
UR - http://eudml.org/doc/279419
ER -