BK:Section 3

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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]A[/math] in [math]\mathbb{F}_3^n[/math] of density about [math]1/n[/math] either has a `good' density increment on a subspace of codimension [math]d[/math], or else the [math](1/n)[/math]-large spectrum of [math]A[/math] intersects any [math]d[/math]-dimensional subspace in at most about [math]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]A[/math] be a subset of [math]\mathbb{F}_3^n[/math] with density [math]\alpha[/math], and let [math]\delta \gt 0[/math] and [math]0 \leq \eta \leq 1[/math] be parameters. Set [math]\Delta = \{ \gamma \in \widehat{G} : | \widehat{1_A}(\gamma) | \geq \delta \alpha \} \setminus \{0\}[/math]. Then
  1. either there is a subspace of [math]\mathbb{F}_3^n[/math] of codimension [math]d[/math] on which [math]A[/math] has density at least [math]\alpha(1 + \eta)[/math]
  2. or [math]|\Delta \cap W| \leq \eta \delta^{-2}[/math] for each [math]d[/math]-dimensional subspace [math]W \leq \widehat{\mathbb{F}_3^n}[/math].

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

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

and hence

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

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


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


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

Furthermore, since

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

defining [math]V^-[/math] similarly and combining the trivial estimate

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

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

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

Combining these sum estimates gives

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

and hence

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

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

[math]|\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]\Delta[/math] from Parseval alone
  • Statement of Chang's theorem
  • Relation to Lemma 2.8 in Sanders's paper