AN INVERSE THEOREM FOR THE GOWERS U4-NORM

GREEN, BEN ; TAO, TERENCE ; ZIEGLER, TAMAR

Glasgow mathematical journal, Tome 53 (2011) no. 1, pp. 1-50

Résumé

We prove the so-called inverse conjecture for the Gowers Us+1-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we show that if f : [N] → C is a function with |f(n)| ≤ 1 for all n and ‖f‖U4 ≥ δ then there is a bounded complexity 3-step nilsequence F(g(n)Γ) which correlates with f. The approach seems to generalise so as to prove the inverse conjecture for s ≥ 4 as well, and a longer paper will follow concerning this.By combining the main result of the present paper with several previous results of the first two authors one obtains the generalised Hardy–Littlewood prime-tuples conjecture for any linear system of complexity at most 3. In particular, we have an asymptotic for the number of 5-term arithmetic progressions p1 < p2 < p3 < p4 < p5 ≤ N of primes.

Détail

DOI : 10.1017/S0017089510000546

Mots-clés : Primary: 11B30, Secondary: 37A17

Bibliographie

[1] 1.Bergelson, V., Tao, T. C. and Ziegler, T., An inverse theorem for uniformity seminorms associated with the action of Fω, Geom. Funct. Anal. 19 (6) (2010), 1539–1596. Google Scholar | DOI

[2] 2.Bogolyubov, N. N., Sur quelques propriétés arithmétiques des presque-périodes, Ann. Chaire Math. Phys. Kiev 4 (1939), 185–194. Google Scholar

[3] 3.Bourbaki, N., Lie groups and Lie algebras. Chaps. 1–3 (translated from French), in Elements of mathematics (Springer-Verlag, Berlin, 1998), xviii + 450 pp. Google Scholar

[4] 4.Bourgain, J., On arithmetic progressions in sums of sets of integers, in A tribute to Paul Erdȍs (Cambridge University Press, Cambridge, UK, 1990), 105–109. Google Scholar | DOI

[5] 5.Bourgain, J., On triples in arithmetic progression, Geom. Funct. Anal. 9 (5) (1999), 968–984. Google Scholar | DOI

[6] 6.Furstenberg, H. and Weiss, B., A mean ergodic theorem for , in Convergence in ergodic theory and probability (Columbus, OH, 1993), Ohio State University Mathematical Research Institute Publications, vol. 5 (de Gruyter, Berlin, 1996), 193–227. Google Scholar

[7] 7.Gowers, W. T., A new proof of Szemerédi's theorem for progressions of length four, GAFA 8 (3) (1998), 529–551. Google Scholar

[8] 8.Gowers, W. T., A new proof of Szemerédi's theorem, GAFA 11 (2001), 465–588. Google Scholar

[9] 9.Green, B. J., Arithmetic progressions in sumsets, GAFA 12 (3) (2002), 584–597. Google Scholar

[10] 10.Green, B. J., Generalising the Hardy–Littlewood method for primes, in International Congress of Mathematicians, vol. II (European Mathematical Society, Zurich, 2006), 373–399. Google Scholar

[11] 11.Green, B. J. and Tao, T. C., An inverse theorem for the Gowers U 3-norm, with applications, Proc. Edinburgh Math. Soc. 51 (1) (2008), 71–153. Google Scholar | DOI

[12] 12.Green, B. J. and Tao, T. C., Quadratic uniformity of the Möbius function, Ann. l'Inst. Fourier (Grenoble) 58 (6) (2008), 1863–1935. Google Scholar

[13] 13.Green, B. J. and Tao, T. C., Linear equations in primes, Ann. Math. (in press). Google Scholar

[14] 14.Green, B. J. and Tao, T. C., The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. Math. (in press). Google Scholar

[15] 15.Green, B. J. and Tao, T. C., The Möbius function is strongly orthogonal to nilsequences, Ann. Math. (in press). Google Scholar

[16] 16.Green, B. J. and Tao, T. C., An arithmetic regularity lemma, associated counting lemma, and applications, in Proceedings of the conference in honour of the 70th birthday of Endre Szemerédi (in press). Google Scholar

[17] 17.Green, B. J., Tao, T. C. and Ziegler, T., An inverse theorem for the Gowers norms (manuscript submitted for publication). Google Scholar

[18] 18.Host, B. and Kra, B., Nonconventional ergodic averages and nilmanifolds, Ann. Math. (2) 161 (1) (2005), 397–488. Google Scholar

[19] 19.Hrushovski, E., Totally categorical structures, Trans. Amer. Math. Soc. 313 (1) (1989), 131–159. Google Scholar

[20] 20.Leibman, A., Polynomial sequences in groups, J. Algebra 201 (1998), 189–206. Google Scholar | DOI

[21] 21.Leibman, A., Pointwise convergence of ergodic averages of polynomial sequences of translations on a nilmanifold, Ergodic Theory Dyn. Syst. 25 (1) (2005), 201–213. Google Scholar

[22] 22.Leibman, A., A canonical form and the distribution of values of generalised polynomials, Israel J. Math. Google Scholar

[23] 23.Lev, V., Optimal representations by sumsets and subset sums, J. Number Theory 62 (1) (1997), 127–143. Google Scholar | DOI

[24] 24.Sárközy, A., Finite addition theorems, I, J. Number Theory 32 (1989), 114–130. Google Scholar | DOI

[25] 25.Tao, T. C. and Vu, V., Additive combinatorics, in Cambridge studies in advanced mathematics, vol. 105 (Cambridge University Press, Cambridge, UK, 2006). Google Scholar

[26] 26.Tao, T. C. and Ziegler, T., The inverse conjecture for the Gowers norms over finite fields via the correspondence principle, Analysis PDE (in press). Google Scholar

[27] 27.Vaughan, R. C., The Hardy–Littlewood method, in Cambridge tracts in mathematics, vol. 80 (Cambridge University Press, New York, 1981). Google Scholar

[28] 28.Ziegler, T., Universal characteristic factors and Furstenberg averages, J. Amer. Math. Soc. 20 (2007), 53–97. Google Scholar