An inverse theorem for the Gowers U^4 norm

Details An inverse theorem for the Gowers U^4 norm An inverse theorem for the Gowers U^4 norm

2009-11-30

arxiv.org/abs/0911.5681v3

Glasgow Mathematical Journal (2011), 53 : pp 1-50

Mathematics

Number Theory

49 pages, to appear in Glasgow J. Math. Fixed a problem with the file (the paper appeared in duplicate)

Scientific paper

10.1017/S0017089510000546

We prove the so-called inverse conjecture for the Gowers U^{s+1}-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we establish that if f : [N] -> C is a function with |f(n)| <= 1 for all n and || f ||_{U^4} >= \delta then there is a bounded complexity 3-step nilsequence F(g(n)\Gamma) 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 this with several previous papers 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 p_1 < p_2 < p_3 < p_4 < p_5 <= N of primes.

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