Translation invariant quadratic forms in $8$ variables and dense sets of primes

Source abstract

Let $Q$ be a translation invariant indefinite quadratic form in $8$ variables. Let $\mathcal{P}$ be the set of all primes. Let $\mathcal{A}\subset\mathcal{P}\cap\{1,\dots,X\}$. By developing the recent work of Green, we prove that subject to a mild condition on the form $Q$ there exist pairwise distinct primes $p_{1},\dots,p_{8}\in \mathcal{A}$ satisfying $Q(p_{1},\dots,p_{8})=0$ provided that $$\frac{|\mathcal{A}|}{X/\log X}\gg_{Q}(\log\log X)^{-c}$$ for some absolute constant $c>0$.