Translation invariant quadratic forms and dense sets of primes

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