The ternary Goldbach problem with primes in positive density sets

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})\...

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Bibliographic Details
Main Author Shen, Quanli
Format Journal Article
LanguageEnglish
Published 27.02.2016
Subjects
Mathematics - Number Theory
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Summary: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.
DOI:10.48550/arxiv.1603.00004