The value of additive forms at prime arguments
Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 77-91.

Soit f(𝐩) une forme de degré k en s variables p 1 ,p 2 ,⋯,p s , où les p i sont des nombres premiers. On suppose que les coefficients λ i de f sont réels, et qu’au moins l’un des rapports λ i /λ j est irrationnel et algébrique. Si s est assez grand alors f prend des valeurs proches de tous les termes d’une suite arbitraire de nombres bien-espacés. Cela généralise des travaux antérieurs de Brüdern, Cook et Perelli (formes linéaires), et Cook et Fox (formes quadratiques). La preuve de ce résultat dépend du Lemme de Hua et, pour k≥6 des améliorations dues à Heath-Brown de ce lemme.

Let f(𝐩) be an additive form of degree k with s prime variables p 1 ,p 2 ,⋯,p s . Suppose that f has real coefficients λ i with at least one ratio λ i /λ j algebraic and irrational. If s is large enough then f takes values close to almost all members of any well-spaced sequence. This complements earlier work of Brüdern, Cook and Perelli (linear forms) and Cook and Fox (quadratic forms). The result is based on Hua’s Lemma and, for k≥6, Heath-Brown’s improvement on Hua’s Lemma.

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Roger J. Cook. The value of additive forms at prime arguments. Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 77-91. https://jtnb.centre-mersenne.org/item/JTNB_2001__13_1_77_0/

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