# Sums of positive density subsets of the primes

Acta Arithmetica (2013)

- Volume: 159, Issue: 3, page 201-225
- ISSN: 0065-1036

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topKaisa 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 -

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