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The values of ternary quadratic forms at prime arguments. (English) Zbl 1107.11043
A set \(\mathcal V\) of positive reals is called well-spaced if there is a constant \(c > 0\) such that for any two distinct elements \(u, v \in \mathcal V\) we have \(| u - v| > c\). Let \(\mathcal V\) be a well-spaced set such that \(\#(\mathcal V \cap [0, X]) \gg X^{1 - \epsilon}\) for any fixed \(\epsilon > 0\). Given non-zero reals \(\lambda_1, \lambda_2, \lambda_3\), let \(E(\mathcal V, X, \delta)\) denote the number of \(v \in (\mathcal V \cap [0, X])\) such that the inequality \[ \left| \lambda_1p_1^2 + \lambda_2p_2^2 + \lambda_3p_3^2 - v \right| < v^{-\delta} \] has no solution in primes \(p_1, p_2, p_3\). If \(\lambda_1/\lambda_2\) is irrational, one expects \(E(\mathcal V, X, \delta)\) to be small (or even bounded). In this paper, the author shows that there is a sequence \(X_j \to \infty\) such that \[ E(\mathcal V, X_j, \delta) \ll X_j^{7/8 + 2\delta + \epsilon} \] for any fixed \(\epsilon > 0\). Furthermore, when the denominators \(q_j\) of the convergents to the continued fraction for \(\lambda_1/\lambda_2\) satisfy \(q_{j + 1} \ll q_j^{1 + \epsilon}\), he proves that \[ E(\mathcal V, X, \delta) \ll X^{6/7 + 2\delta + \epsilon} \eqno{(1)} \] for any fixed \(\epsilon > 0\) and all \(X \geq X_0(\epsilon)\). These results improve on earlier work by R. Cook and A. Fox [Mathematika 48, 137–149 (2001; Zbl 1035.11010)] where (1) is established with the exponent \(11/12\) in place of \(6/7\). The proof uses the Davenport–Heilbronn version of the circle method, estimates for Weyl sums over primes and almost primes, and a sieve method.

11P32 Goldbach-type theorems; other additive questions involving primes
11D75 Diophantine inequalities
11N36 Applications of sieve methods
Zbl 1035.11010
Full Text: DOI
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