General systems of linear forms: equidistribution and true complexity

Hamed Hatami, Pooya Hatami, Shachar Lovett

Published 2014-03-30, updated 2014-05-07Version 2

The densities of small linear structures (such as arithmetic progressions) in subsets of Abelian groups can be expressed as certain analytic averages involving linear forms. Higher-order Fourier analysis examines such averages by approximating the indicator function of a subset by a function of bounded number of polynomials. Then, to approximate the average, it suffices to know the joint distribution of the polynomials applied to the linear forms. We prove a near-equidistribution theorem that describes these distributions for the group $\mathbb{F}_p^n$ when $p$ is a fixed prime. This fundamental fact is equivalent to a strong near-orthogonality statement regarding the higher-order characters, and was previously known only under various extra assumptions about the linear forms. As an application of our near-equidistribution theorem, we settle a conjecture of Gowers and Wolf on the true complexity of systems of linear forms for the group $\mathbb{F}_p^n$.