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The value of additive forms at prime arguments. (English) Zbl 1047.11095
Let \(k\) and \(s\) be integers satisfying either \(k\geq3\) and \(s=2^{k-1}+1\), or \(k\geq6\) and \(s=7\cdot2^{k-4}+1\), let \(\lambda_1,\ldots,\lambda_s\) and \(\delta\) be positive real numbers, and suppose that \(\lambda_1/\lambda_2\) is irrational and algebraic. Write \(E(X)\) for the measure of the set of positive real numbers \(v\leq X\) such that the inequality \[ | \lambda_1p_1+\cdots+\lambda_sp_s-v| <v^{-\delta} \] has no solution in primes \(p_1,\ldots,p_s\). Then the main theorem of this paper asserts that for any \(\varepsilon>0\), one has \(E(X)\ll X^{1-2^{3-2k}(3k)^{-1}+2\delta+\varepsilon}\). This work may be regarded as a generalisation of that of R. J. Cook and A. Fox [Mathematika 48, No. 1–2, 137–149 (2001; Zbl 1035.11010)], in which the above conclusion was shown for \(k=2\). For the case \(k=1\), see J. Brüdern, R. J. Cook and A. Perelli [Sieve methods, exponential sums, and their applications in number theory, Lond. Math. Soc. Lect. Note Ser. 237, 87–100 (1997; Zbl 0924.11085)].
The proof is based on the method of Davenport and Heilbronn, and the minor arc integral is estimated by means of Hua’s lemma, Heath-Brown’s improvement of the latter lemma, and Harman’s estimate for exponential sums over primes.

MSC:
11P32 Goldbach-type theorems; other additive questions involving primes
11D75 Diophantine inequalities
11P55 Applications of the Hardy-Littlewood method
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