Translation invariant quadratic forms and dense sets of primes

Abstract

Let $f(x_1,\dots ,x_s)$ be a translation invariant indefinite quadratic form of integer coefficients with $s\ge 10$. Let $\mathcal{A}\subseteq \mathcal{P}\cap \{1,2,\dots ,X\}$. Let $X$ be sufficiently large. Subject to a rank condition, we prove that there exist distinct primes $p_1,\dots ,p_s\in \mathcal{A}$ such that $f(p_1,\dots ,p_s)=0$ as soon as $|\mathcal{A}|\ge \frac{X}{\log X}(\log \log X{)}^{-\frac{1}{80}}.$

References in corpus (2)

• Discorrelation between primes in short intervals and polynomial phases
• Some refinements for translation invariant quadratic forms in dense sets