An inverse theorem for the Gowers U^4 norm

Ben Green
University of Cambridge

The Gowers norms U^2, U^3, U^4 have been a central object of study in additive combinatorics for several years. The inverse conjectures for the Gowers norms make a certain prediction about precisely when the U^s norm of a function is large. Tao and I have reduced the proof of a conjecture of Hardy and Littlewood concerning configurations of prime numbers to the task of establishing these conjectures.

An inverse theorem for the U^2 norm can be established easily using Fourier analysis. Tao and I proved an inverse theoren for the U^3 norm a few years ago. I will talk about a recent proof (joint work with Terry Tao and Tamar
Ziegler) of an inverse theorem for the Gowers U^4 norm. The approach appears to generalise to establish the inverse conjectures in general.

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