AN INVERSE THEOREM FOR THE GOWERSU4-NORM

Abstract

AbstractWe prove the so-calledinverse conjecture for the Gowers Us+1-normin the cases= 3 (the casess< 3 being established in previous literature). That is, we show that iff: [N] → ℂ is a function with |f(n)| ≤ 1 for allnand ‖f‖U4≥ δ then there is a bounded complexity 3-step nilsequenceF(g(n)Γ) which correlates withf. The approach seems to generalise so as to prove the inverse conjecture fors≥ 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 progressionsp1<p2<p3<p4<p5≤Nof primes.