Sums of positive density subsets of the primes

Kaisa Matomäki

Acta Arithmetica, Tome 161 (2013), p. 201-225 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 2013-01-01

Zbl 06177628

EUDML-ID : urn:eudml:doc:278962