Also appears in collection : 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.
By Artem Chernikov
By Martin Bays
By Micha Sharir
By David Evans
By Pierre Simon
By Timothy Gowers
By Caroline Terry
By Saugata Basu
By Jan Hubička
By Gérard Ben Arous
By Giulio Biroli
By Giulio Biroli
By Gérard Ben Arous
By Gérard Ben Arous
