General systems of linear forms: equidistribution and true complexity

 Computer Science/Discrete Mathematics Seminar II Topic: General systems of linear forms: equidistribution and true complexity Speaker: Pooya Hatami Affiliation: Member, School of Mathematics Date: Tuesday, November 24 Time/Room: 10:30am - 12:30pm/S-101 Video Link: https://video.ias.edu/csdm/2015/1124-Hatami

Higher-order Fourier analysis is a powerful tool that can be used to analyse the densities of linear systems (such as arithmetic progressions) in subsets of Abelian groups. We are interested in the group $\mathbb{F}_p^n$, for fixed $p$ and large $n$, where it is known that analysing these averages reduces to understanding the joint distribution of a family of sufficiently pseudorandom (formally, high rank) nonclassical polynomials applied to the corresponding system of linear forms. In this work, we give a complete characterization for these distributions for arbitrary systems of linear forms. This extends previous works which accomplished this in some special cases. As an application, we resolve a conjecture of Gowers and Wolf on the true complexity of linear systems. Our proof deviates from that of the previously known special cases and requires several new ingredients. One of which, which may be of independent interest, is a new theory of homogeneous nonclassical polynomials.