Translation invariant quadratic forms and dense sets of primes

Zhao, Lilu

doi:10.1090/tran/8530

Translation invariant quadratic forms and dense sets of primes
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Trans. Amer. Math. Soc. 375 (2022), 725-752 Request permission

Abstract:

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 $|\mathcal {A}|{\,\geqslant \,} \frac {X}{\log X} (\log \log X)^{-\frac {1}{80}}.$

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