ADDITIVE ENERGY OF DENSE SETS OF PRIMES AND MONOCHROMATIC SUMS

International audience; 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.