The ternary Goldbach problem with primes in positive density sets

• Published in: arXiv.org
• Main Author: Shen, Quanli
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 27.02.2016
• Subjects: Density

Let \(\mathcal{P}\) denote the set of all primes. \(P_{1},P_{2},P_{3}\) are three subsets of \(\mathcal{P}\). Let \(\underline{\delta}(P_{i})\) \((i=1,2,3)\) denote the lower density of \(P_{i}\) in \(\mathcal{P}\), respectively. It is proved that if \(\underline{\delta}(P_{1})>5/8\), \(\underline{\delta}(P_{2})\geq5/8\), and \(\underline{\delta}(P_{3})\geq5/8\), then for every sufficiently large odd integer n, there exist \(p_{i} \in P_{i}\) such that \(n=p_{1}+p_{2}+p_{3}\). The condition is the best possible.