General systems of linear forms: Equidistribution and true complexity

Abstract

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 nearequidistribution theorem that describes these distributions for the group F n p 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 F n p .

A counting lemma for binary matroids and applications to extremal problems

“…Here we will address the technical issues that arise in working with the first two parts. In the sections below, we largely quote the terminology and notation used in [3] and [11].…”

“…For ease of notation in the proof that follows, we will introduce the concept of a linear form, as used in [3] and [11].…”

“…Our main result in this paper is a corresponding Counting Lemma for matroids, Theorem 3.13, which we develop by building on work in [3] and [11]. The case of the Counting Lemma for affine matroids (i.e.…”

Stability and exact Turán numbers for matroids

“…This paper deals with Turán-type problems in the arithmetic setting of subsets of F n 2 , where we fix a set N ⊆ F k 2 and consider, for n much larger than k, the size of a set M ⊆ F n 2 that does not contain any subset that is the image of N under an injective linear map ϕ : F k 2 → F n 2 . This is analogous to excluding a fixed subgraph H from a graph G. Such problems have been considered both in the language of arithmetic combinatorics [1,9,10] and equivalently matroid theory [6,7,15]; here we will use the term 'matroid' for brevity to describe the relevant notions of containment and isomorphism, as well as to highlight the strong analogies with graph theory, and to describe the 'host' object M and the 'system of linear forms' being excluded in a unified way. While they are stated combinatorially, all of our results depend on a Fourieranalytic regularity lemma of Hatami et al [10] and a new associated counting lemma due to the second author [11].…”

“…Our main results still hold if 'matroid' is just read as 'simple binary matroid'. In the other direction, stating that a matroid M does not contain N is equivalent to insisting that M does not contain a nondegenerate copy of some 'system of linear forms' in the language of [10]. If N = {w 1 , .…”