Appears in collections : 2018 - T1 - WS1 - Model theory and combinatorics, Fields medallists - 1998

The U⁴ norm is one of a sequence of norms that measure ever stronger forms of quasirandomness. The structure of bounded functions whose Uᵏ norms are within a constant of being as large as possible has been the subject of a lot of research over the last twenty years, and has applications to results such as Szemerédi’s theorem and the Green–Tao theorem. Qualitatively speaking, there is now a complete description of such functions when they are defined on ? n p (a result of Bergelson, Tao and Ziegler) and ℤN (a result of Green, Tao and Ziegler). I shall describe recent work with Luka Milićević in which we obtain quantitative bounds for the first case where these were not known, namely for the U⁴ norm and for functions defined on ?^n_p.