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Sums of positive density subsets of the primes

Volume 159 / 2013

Kaisa Matomäki Acta Arithmetica 159 (2013), 201-225 MSC: Primary 11B30, 11P32. DOI: 10.4064/aa159-3-1

Abstract

We show that if $A$ and $B$ are subsets of the primes with positive relative lower densities $\alpha$ and $\beta$, then the lower density of $A+B$ in the natural numbers is at least $(1-o(1))\alpha/(e^{\gamma}\log \log (1/\beta))$, 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 $\mathbb Z_m^\ast$ using techniques of Green and Green–Tao. Concerning this new problem we show that, for any square-free $m$ and any $A, B \subseteq \mathbb Z_m^\ast$ of densities $\alpha$ and $\beta$, the density of $A+B$ in $\mathbb Z_m$ is at least $(1-o(1))\alpha/(e^{\gamma} \log \log (1/\beta))$, which is asymptotically best possible when $m$ is a product of small primes. We also discuss an inverse question.

Authors

  • Kaisa MatomäkiDepartment of Mathematics
    University of Turku
    20014 Turku, Finland
    e-mail

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