BK:Section 3

Parent page: Improving the bounds for Roth's theorem

One of the take-away results from Section 3 of the Bateman-Katz paper is Proposition 3.1, an important part of 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. We shall say later on why this is significant.

Here is the precise result, stated in slightly different terms to the paper in order to illustrate how it relates to other results. For a subspace [math]\displaystyle{ V \leq \mathbb{F}_3^n }[/math] we write

[math]\displaystyle{ V^{\perp} = \{ \gamma \in \widehat{\mathbb{F}_3^n} : \gamma(x) = 1 \ \forall x \in V \} }[/math]

for its annihilator (cf. the section on Bohr sets).

Proposition 1 Let [math]\displaystyle{ A \subset \mathbb{F}_3^n }[/math] be a set with density [math]\displaystyle{ \alpha }[/math], and let [math]\displaystyle{ 0 \leq \delta, \eta \leq 1 }[/math] be parameters. Set

[math]\displaystyle{ \Delta = \{ \gamma \in \widehat{G} : | \widehat{1_A}(\gamma) | \geq \delta \alpha \} \setminus \{ 0_{\widehat{\mathbb{F}_3^n}} \} }[/math].

Suppose [math]\displaystyle{ V \leq \mathbb{F}_3^n }[/math] be a subspace. Then

• either [math]\displaystyle{ A }[/math] has density at least [math]\displaystyle{ \alpha(1 + \eta) }[/math] on [math]\displaystyle{ V }[/math],
• or [math]\displaystyle{ |\Delta \cap V^{\perp}| \leq 3\eta \delta^{-2} }[/math]; in fact [math]\displaystyle{ \sum_{\gamma \in V^{\perp}} |\widehat{(1_A - \alpha)}(\gamma)|^2 \leq 3\eta \alpha^2 }[/math].

Proof: Let us write [math]\displaystyle{ \mu_V = \frac{|\mathbb{F}_3^n|}{|V|}1_V }[/math] for the indicator function of [math]\displaystyle{ V }[/math] normalized so that [math]\displaystyle{ \mathbb{E}_x \mu_V(x) = 1 }[/math]. If

[math]\displaystyle{ 1_A*\mu_V(x) \gt \alpha(1 + \eta) }[/math]

for some [math]\displaystyle{ x \in \mathbb{F}_3^n }[/math] then we are in the first case, so let us assume that [math]\displaystyle{ 1_A*\mu_V \leq \alpha(1+\eta) }[/math]. Write [math]\displaystyle{ f = 1_A - \alpha }[/math] for the balanced function of [math]\displaystyle{ A }[/math]. Then

[math]\displaystyle{ | \Delta \cap V^{\perp} | \delta^2 \alpha^2 \leq \sum_{\gamma \in V^{\perp}} |\widehat{f}(\gamma)|^2 = \sum_{\gamma \in \widehat{\mathbb{F}_3^n}} |\widehat{f}(\gamma)|^2 |\widehat{\mu_V}(\gamma)|^2. }[/math]

By Parseval's identity, this equals

[math]\displaystyle{ \mathbb{E}_{x \in \mathbb{F}_3^n} f*\mu_V(x)^2 = \mathbb{E}_{x \in \mathbb{F}_3^n} 1_A*\mu_V(x)^2 - \alpha^2 \leq \alpha^2(2\eta + \eta^2), }[/math]

which proves the result.

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