Sums of positive density subsets of the primes

Kaisa Matomäki¹

¹ Department of Mathematics University of Turku 20014 Turku, Finland

Acta Arithmetica, Tome 159 (2013) no. 3, pp. 201-225

Résumé

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.

Détail

DOI : 10.4064/aa159-3-1

Keywords: subsets primes positive relative lower densities alpha beta lower density natural numbers least o alpha gamma log log beta which asymptotically best possible improves results ramar ruzsa chipeniuk hamel latter work problem reduced similar problem subsets mathbb ast using techniques green green tao concerning problem square free subseteq mathbb ast densities alpha beta density mathbb least o alpha gamma log log beta which asymptotically best possible product small primes discuss inverse question