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].

Proof Choose a subspace [math]\displaystyle{ H }[/math] such that [math]\displaystyle{ W }[/math] is the annihilator of [math]\displaystyle{ H }[/math], and let [math]\displaystyle{ V }[/math] be a subspace transverse to [math]\displaystyle{ H }[/math]. Then for any [math]\displaystyle{ \gamma\neq0\in W }[/math],

[math]\displaystyle{ \widehat{1_A}(\gamma)=3^{-n}\sum_{v\in V}(| A\cap(H+v)|-3^{-d}| A|)\gamma(v) }[/math]

and hence

[math]\displaystyle{ \sum_{\gamma\neq0\in W}|\widehat{1_A}(\gamma)|^2=3^{d-2n}\sum_{v\in V}(| A\cap(H+v)|-3^{-d}| A|)^2. }[/math]

If we let [math]\displaystyle{ V^+ }[/math] be the subset of [math]\displaystyle{ V }[/math] for which each of the squared summands is positive, then either [math]\displaystyle{ A }[/math] has the required density increment on a translate of [math]\displaystyle{ H }[/math] (which has codimension [math]\displaystyle{ d }[/math]), or

[math]\displaystyle{ || A\cap(H+v)|-3^{-d}| A||\ll 3^{-d}| A|\eta. }[/math]

Hence

[math]\displaystyle{ \sum_{v\in V^+}|| A\cap(H+v)|-3^{-d}| A||\ll| A|\eta }[/math]

and

[math]\displaystyle{ \sum_{v\in V^+}|| A\cap(H+v)|-3^{-d}| A||^2\ll 3^{-d}| A|^2\eta^2. }[/math]

Furthermore, since

[math]\displaystyle{ \sum_{v\in V}|| A\cap(H+v)|-3^{-d}| A||=0 }[/math]

defining [math]\displaystyle{ V^- }[/math] similarly and combining the trivial estimate

[math]\displaystyle{ || A\cap(H+v)|-3^{-d}| A||\leq3^{-d}| A| }[/math]

for [math]\displaystyle{ v\in V^- }[/math] with the above gives

[math]\displaystyle{ \sum_{v\in V^-}|| A\cap(H+v)|-3^{-d}| A||^2\ll3^{-d}| A|^2\eta. }[/math]

Combining these sum estimates gives

[math]\displaystyle{ \sum_{v\in V}|| A\cap(H+v)|-3^{-d}| A||^2\ll3^{-d}| A|^2\eta }[/math]

and hence

[math]\displaystyle{ \sum_{\gamma\neq0\in W}|\widehat{1_A}(\gamma)|^2\ll \alpha^2\eta. }[/math]

Recalling the definition of [math]\displaystyle{ \Delta }[/math], we have

[math]\displaystyle{ |\Delta\cap W|\delta^2\alpha^2\ll\sum_{\gamma\in\Delta\cap W}|\widehat{1_A}(\gamma)|^2\ll\alpha^2\eta. }[/math]

To be added:

• 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