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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