An inverse theorem for the Gowers U4-norm

Ben Green*, Terence Tao, Tamar Ziegler

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

21 Scopus citations


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 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)) 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.

Original languageAmerican English
Pages (from-to)1-50
Number of pages50
JournalGlasgow Mathematical Journal
Issue number1
StatePublished - Jan 2011
Externally publishedYes


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