ADDITIVE ENERGY OF DENSE SETS OF PRIMES AND MONOCHROMATIC SUMS

Ramana, D; Ramaré, Olivier

Type de document :

Compte-rendu et recension critique d'ouvrage

Titre :

ADDITIVE ENERGY OF DENSE SETS OF PRIMES AND MONOCHROMATIC SUMS

Auteur(s) :

Ramana, D [Auteur]
Ramaré, Olivier [Auteur]
Centre National de la Recherche Scientifique [CNRS]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Titre de la revue :

Israel Journal of Mathematics

Pagination :

955-974

Éditeur :

Springer

Date de publication :

2014

ISSN :

0021-2172

Discipline(s) HAL :

Mathématiques [math]/Théorie des nombres [math.NT]

Résumé en anglais : [en]

When $K \ge 1$ is an integer and $S$ is a set of prime numbers in the interval $(N/2 , N ]$ with $|S| \ge\pi^* (N)/K$, where $\pi^* (N)$ is the number of primes in this interval, we obtain an upper bound for the additive ...
Lire la suite >When $K \ge 1$ is an integer and $S$ is a set of prime numbers in the interval $(N/2 , N ]$ with $|S| \ge\pi^* (N)/K$, where $\pi^* (N)$ is the number of primes in this interval, we obtain an upper bound for the additive energy of $S$, which is the number of quadruples $(x_1 , x_2 , x_3 , x_4)$ in $S^4$ satisfying $x_1 + x_2 = x_3 + x_4$. We obtain this bound by a variant of a method of Ramaré and I. Ruzsa. Taken together with an argument due to N. Hegyvári and F. Hennecart this bound implies that when the sequence of prime numbers is coloured with $K$ colours, every sufficiently large integer can be written as a sum of no more than $CK \log \log 4K$ prime numbers, all of the same colour, where $C$ is an absolute constant. This assertion is optimal upto the value of C and answers a question of A. Sárközy.Lire moins >

Langue :

Anglais

Vulgarisation :

Non

Collections :

Laboratoire Paul Painlevé - UMR 8524

Source :

Harvested from HAL

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