The value of additive forms at prime arguments

Cook, Roger J.

Journal de théorie des nombres de Bordeaux, Tome 13 (2001) no. 1, pp. 77-91.

Soit $f\left(𝐩\right)$ une forme de degré $k$ en $s$ variables $p_1,p_2,\cdots ,p_s$, où les $p_i$ sont des nombres premiers. On suppose que les coefficients ${\lambda }_i$ de $f$ sont réels, et qu’au moins l’un des rapports ${\lambda }_i/{\lambda }_j$ 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 $k\ge 6$ des améliorations dues à Heath-Brown de ce lemme.

Let $f\left(𝐩\right)$ be an additive form of degree $k$ with $s$ prime variables $p_1,p_2,\cdots ,p_s$. Suppose that $f$ has real coefficients ${\lambda }_i$ with at least one ratio ${\lambda }_i/{\lambda }_j$ 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 $k\ge 6$, Heath-Brown’s improvement on Hua’s Lemma.

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