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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
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Abstract

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

References

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