BK:Section 3

One of the take-away results from Section 3 of the Bateman-Katz paper is Proposition 3.1, which is in some places referred to as the "nd-estimate". The rough reason for this terminology is that it says that a set [math]\displaystyle{ A }[/math] in [math]\displaystyle{ \mathbb{F}_3^n }[/math] of density about [math]\displaystyle{ 1/n }[/math] either has a `good' density increment on a subspace of codimension [math]\displaystyle{ d }[/math], or else the [math]\displaystyle{ (1/n) }[/math]-large spectrum of [math]\displaystyle{ A }[/math] intersects any [math]\displaystyle{ d }[/math]-dimensional subspace in at most about [math]\displaystyle{ nd }[/math] points.

Here is the precise result, stated in slightly different terms to the paper in order to illustrate how it relates to other results.

Proposition 1 Let [math]\displaystyle{ A }[/math] be a subset of [math]\displaystyle{ \mathbb{F}_3^n }[/math] with density [math]\displaystyle{ \alpha }[/math], and let [math]\displaystyle{ \delta \gt 0 }[/math] and [math]\displaystyle{ 0 \leq \eta \leq 1 }[/math] be parameters. Set [math]\displaystyle{ \Delta = \{ \gamma \in \widehat{G} : | \widehat{1_A}(\gamma) | \geq \delta \alpha \} \setminus \{0\} }[/math]. Then

1. either there is a subspace of [math]\displaystyle{ \mathbb{F}_3^n }[/math] of codimension [math]\displaystyle{ d }[/math] on which [math]\displaystyle{ A }[/math] has density at least [math]\displaystyle{ \alpha(1 + \eta) }[/math]
2. or [math]\displaystyle{ |\Delta \cap W| \leq \eta \delta^{-2} }[/math] for each [math]\displaystyle{ d }[/math]-dimensional subspace [math]\displaystyle{ W \leq \widehat{\mathbb{F}_3^n} }[/math].

To be added:

• Proof
• Statement of size bound on [math]\displaystyle{ \Delta }[/math] from Parseval alone
• Statement of Chang's theorem
• Relation to Lemma 2.8 in Sanders's paper