An inverse theorem for the Gowers Us+1[N]-norm

Ben Green*, Terence Tao, Tamar Ziegler

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

137 Scopus citations


We prove the inverse conjecture for the Gowers Us+1[N]-norm for all s ≥ 1; this is new for s ≥ 4. More precisely, we establish that if f: [N] → [-1; 1] is a function with ||f||Us+1[N] ≥ δ, then there is a bounded complexitys-step nilsequence F(g(n)Γ) that correlates with f, where the bounds on the complexity and correlation depend only on s and δ. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity.

Original languageAmerican English
Pages (from-to)1231-1372
Number of pages142
JournalAnnals of Mathematics
Issue number2
StatePublished - Sep 2012
Externally publishedYes


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