Hostname: page-component-7dd5485656-wjfd9 Total loading time: 0 Render date: 2025-10-30T20:24:08.708Z Has data issue: false hasContentIssue false
  • English
  • Français

AN INVERSE THEOREM FOR THE GOWERS U4-NORM

Published online by Cambridge University Press:  25 August 2010

BEN GREEN ,
TERENCE TAO  and
TAMAR ZIEGLER
BEN GREEN
Affiliation:
Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK e-mail: b.j.green@dpmms.cam.ac.uk
TERENCE TAO
Affiliation:
UCLA Department of Mathematics, Los Angeles, CA 90095-1555, USA e-mail: tao@math.ucla.edu
TAMAR ZIEGLER
Affiliation:
Department of Mathematics, Technion - Israel Institute of Technology, Haifa 32000, Israel e-mail: tamarzr@tx.technion.ac.il
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Keywords

Primary: 11B30Secondary: 37A17

Information

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 53 , Issue 1 , January 2011 , pp. 1 - 50
Copyright
Copyright © Glasgow Mathematical Journal Trust 2010

References

REFERENCES

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.CrossRefGoogle Scholar
2.Bogolyubov, N. N., Sur quelques propriétés arithmétiques des presque-périodes, Ann. Chaire Math. Phys. Kiev 4 (1939), 185–194.Google Scholar
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.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.CrossRefGoogle Scholar
5.Bourgain, J., On triples in arithmetic progression, Geom. Funct. Anal. 9 (5) (1999), 968–984.CrossRefGoogle Scholar
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.Gowers, W. T., A new proof of Szemerédi's theorem for progressions of length four, GAFA 8 (3) (1998), 529–551.Google Scholar
8.Gowers, W. T., A new proof of Szemerédi's theorem, GAFA 11 (2001), 465–588.Google Scholar
9.Green, B. J., Arithmetic progressions in sumsets, GAFA 12 (3) (2002), 584–597.Google Scholar
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.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.CrossRefGoogle Scholar
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.Green, B. J. and Tao, T. C., Linear equations in primes, Ann. Math. (in press).Google Scholar
14.Green, B. J. and Tao, T. C., The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. Math. (in press).Google Scholar
15.Green, B. J. and Tao, T. C., The Möbius function is strongly orthogonal to nilsequences, Ann. Math. (in press).Google Scholar
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.Green, B. J., Tao, T. C. and Ziegler, T., An inverse theorem for the Gowers norms (manuscript submitted for publication).Google Scholar
18.Host, B. and Kra, B., Nonconventional ergodic averages and nilmanifolds, Ann. Math. (2) 161 (1) (2005), 397–488.Google Scholar
19.Hrushovski, E., Totally categorical structures, Trans. Amer. Math. Soc. 313 (1) (1989), 131–159.Google Scholar
20.Leibman, A., Polynomial sequences in groups, J. Algebra 201 (1998), 189–206.CrossRefGoogle Scholar
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.Leibman, A., A canonical form and the distribution of values of generalised polynomials, Israel J. Math.Google Scholar
23.Lev, V., Optimal representations by sumsets and subset sums, J. Number Theory 62 (1) (1997), 127–143.CrossRefGoogle Scholar
24.Sárközy, A., Finite addition theorems, I, J. Number Theory 32 (1989), 114–130.CrossRefGoogle Scholar
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.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.Vaughan, R. C., The Hardy–Littlewood method, in Cambridge tracts in mathematics, vol. 80 (Cambridge University Press, New York, 1981).Google Scholar
28.Ziegler, T., Universal characteristic factors and Furstenberg averages, J. Amer. Math. Soc. 20 (2007), 53–97.Google Scholar