J. Bo-]-bourgain, On ??(p)-subsets of squares, Israel Journal of Mathematics, vol.9, issue.3, pp.291-311, 1989.
DOI : 10.1007/BF02764948

J. Bourgain, On Triples in Arithmetic Progression, Geometric And Functional Analysis, vol.9, issue.5, pp.968-984, 1999.
DOI : 10.1007/s000390050105

J. Bourgain, . Gr, and B. Green, Roth's theorem on progressions revisited Roth's theorem in the primes, Ann. of Math, issue.2, pp.161-1609, 2005.

B. Green and T. Tao, Restriction theory of the Selberg sieve, with applications, Journal de Th??orie des Nombres de Bordeaux, vol.18, issue.1, pp.147-182, 2006.
DOI : 10.5802/jtnb.538

B. Green and T. Tao, The primes contain arbitrarily long arithmetic progressions, Annals of Mathematics, vol.167, issue.2, pp.481-547, 2008.
DOI : 10.4007/annals.2008.167.481

H. Hr-]-halberstam and H. E. Richert, Sieve Methods, 1974.
DOI : 10.1007/978-1-4613-8227-0_4

. Hb and D. R. Heath-brown, Integer sets containing no arithmetic progressions, J. London Math. Soc, vol.35, issue.2 3, pp.385-394, 1987.

O. Ramaré, On Snirel'man's constant, Ann. Scu. Norm. Pisa, vol.22, pp.645-706, 1995.

K. F. Roth, On certain sets of integers, J. London Math. Soc, vol.28, pp.104-109, 1953.

E. Szemerédi, Integer sets containing no arithmetic progressions, Acta Mathematica Hungarica, vol.28, issue.1-2, pp.56-155, 1990.
DOI : 10.1007/BF01903717

T. Ta-]-tao, T. Tao, and V. Vu, Arithmetic progressions and the primes Additive combinatorics, Collect. Math, vol.Extra, pp.37-88, 1939.

. Va and P. Varnavides, On certain sets of positive density, J. London Math. Soc, vol.34, pp.358-360, 1959.

H. A. Helfgott, United Kingdom E-mail address: h.andres.helfgott@bristol.ac.uk A. de Roton, Institut Elie Cartan, .P. 239, 54506 Vandouevre-l` es-Nancy cedex, France; A. de Roton, Pacific Institute for the Mathematical Sciences