An inverse theorem for the Gowers U^4 norm

TL;DR

This paper proves the inverse theorem for the Gowers U^4 norm, establishing that functions with large U^4 norm correlate with 3-step nilsequences, advancing understanding of higher-order Fourier analysis and prime patterns.

Contribution

It establishes the inverse conjecture for the Gowers U^4 norm, extending previous results to the case s=3 and paving the way for s>=4, with implications for prime number patterns.

Findings

Proved the inverse conjecture for U^4 norm at s=3.

Connected the result to the generalized Hardy-Littlewood prime-tuples conjecture.

Derived asymptotics for 5-term arithmetic progressions of primes.

Abstract

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.