Geodesic
AN INVERSE THEOREM FOR THE GOWERS U4-NORM
Glasgow mathematical journal, Tome 53 (2011) no. 1, pp. 1-50

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DOI

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.
DOI : 10.1017/S0017089510000546
Mots-clés : Primary: 11B30, Secondary: 37A17 
GREEN, BEN; TAO, TERENCE; ZIEGLER, TAMAR. AN INVERSE THEOREM FOR THE GOWERS U4-NORM. Glasgow mathematical journal, Tome 53 (2011) no. 1, pp. 1-50. doi: 10.1017/S0017089510000546
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