Sums of positive density subsets of the primes

Kaisa Matomäki. "Sums of positive density subsets of the primes." Acta Arithmetica 159.3 (2013): 201-225. <http://eudml.org/doc/278962>.

@article{KaisaMatomäki2013,
abstract = {We show that if A and B are subsets of the primes with positive relative lower densities α and β, then the lower density of A+B in the natural numbers is at least $(1-o(1))α/(e^\{γ\} log log (1/β))$, which is asymptotically best possible. This improves results of Ramaré and Ruzsa and of Chipeniuk and Hamel. As in the latter work, the problem is reduced to a similar problem for subsets of $ℤ*_m$ using techniques of Green and Green-Tao. Concerning this new problem we show that, for any square-free m and any $A, B ⊆ ℤ*_m$ of densities α and β, the density of A+B in $ℤ_m$ is at least $(1-o(1))α/(e^\{γ\} log log (1/β))$, which is asymptotically best possible when m is a product of small primes. We also discuss an inverse question.},
author = {Kaisa Matomäki},
journal = {Acta Arithmetica},
keywords = {sum sets; primes; positive density subsets; goldbach type problems},
language = {eng},
number = {3},
pages = {201-225},
title = {Sums of positive density subsets of the primes},
url = {http://eudml.org/doc/278962},
volume = {159},
year = {2013},
}

TY - JOUR
AU - Kaisa Matomäki
TI - Sums of positive density subsets of the primes
JO - Acta Arithmetica
PY - 2013
VL - 159
IS - 3
SP - 201
EP - 225
AB - We show that if A and B are subsets of the primes with positive relative lower densities α and β, then the lower density of A+B in the natural numbers is at least $(1-o(1))α/(e^{γ} log log (1/β))$, which is asymptotically best possible. This improves results of Ramaré and Ruzsa and of Chipeniuk and Hamel. As in the latter work, the problem is reduced to a similar problem for subsets of $ℤ*_m$ using techniques of Green and Green-Tao. Concerning this new problem we show that, for any square-free m and any $A, B ⊆ ℤ*_m$ of densities α and β, the density of A+B in $ℤ_m$ is at least $(1-o(1))α/(e^{γ} log log (1/β))$, which is asymptotically best possible when m is a product of small primes. We also discuss an inverse question.
LA - eng
KW - sum sets; primes; positive density subsets; goldbach type problems
UR - http://eudml.org/doc/278962
ER -