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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.

##### MSC:
 11P32 Goldbach-type theorems; other additive questions involving primes 11D75 Diophantine inequalities 11N36 Applications of sieve methods
##### Keywords:
Diophantine inequalities; ternary forms; circle method
Zbl 1035.11010
Full Text:
##### References:
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