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

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 k6, Heath-Brown’s improvement on Hua’s Lemma.

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 k6 des améliorations dues à Heath-Brown de ce lemme.

@article{JTNB_2001__13_1_77_0,
     author = {Cook, Roger J.},
     title = {The value of additive forms at prime arguments},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {1},
     year = {2001},
     pages = {77-91},
     zbl = {1047.11095},
     mrnumber = {1838071},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2001__13_1_77_0}
}
Cook, Roger J. The value of additive forms at prime arguments. Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 1, pp. 77-91. http://www.numdam.org/item/JTNB_2001__13_1_77_0/

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