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Sums of positive density subsets of the primes. (English) Zbl 1348.11012

Summary: 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 \[ (1-o(1))\frac{\alpha}{e^{\gamma}\log \log (1/\beta)}, \] which is asymptotically best possible. This improves results of O. Ramaré and I. Z. Ruzsa [J. Théor. Nombres Bordx. 13, 559–581 (2001; Zbl 0996.11057)] and of K. Chipeniuk and M. Hamel [J. Lond. Math. Soc., II. Ser. 83, No. 3, 673–690 (2011; Zbl 1215.11019)]. As in the latter work, the problem is reduced to a similar problem for subsets of \(\mathbb Z_m^\ast\) using techniques of B. Green [Ann. Math. (2) 161, No. 3, 1609–1636 (2005; Zbl 1160.11307)] and B. Green and T. Tao [Ann. Math. (2) 167, No. 2, 481–547 (2008; Zbl 1191.11025)]. 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.

MSC:

11B30 Arithmetic combinatorics; higher degree uniformity
11P32 Goldbach-type theorems; other additive questions involving primes
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