Title: The inverse conjecture for the Gowers uniformity norms on [N]
Speaker: Terry Tao
Speaker Info: University of California, Los Angeles
Brief Description:
Special Note:
Abstract:
The Gowers uniformity norms U^{k+1}[N], introduced by Gowers in 2001, is very useful when counting the number of linear patterns (such as arithmetic progressions of length k+2) in a set of integers. The inverse conjecture gives a necessary and sufficient condition for the Gowers norm of a bounded function to be large; roughly speaking, a function n \mapsto f(n) has large U^{k+1}[N] norm if and only if it correlates with a bounded complexity nilsequence n |-> F( g(n) \Gamma ) on a nilmanifold G/\Gamma of step (or degree) k. This generalises the Fourier-analytic case k=1, in which a function has large U^2[N] norm if and only if it correlates with a Fourier phase n |-> e(\xi n); this case is ultimately behind the Fourier-analytic proofs of results such as Roth's theorem, and the inverse conjecture for higher k can similarly be used to give a proof of Szemeredi's theorem. In this talk we outline a proof of the general case of the inverse conjecture, in joint work with Ben Green and Tamar Ziegler. The arguments are based on those of Gowers, with the main new difficulty being that of "integrating" a family of degree k-1 nilsequences to obtain a single degree k nilsequence as an "antiderivative"Date: Friday, February 25, 2011