Difference between revisions of "BK:Section 3"

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# 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>
 
# 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>
 
# 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>.
 
# 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>,
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:<math>\widehat{1_A}(\gamma)=3^{-n}\sum_{v\in V}(| A\cap(H+v)|-3^{-d}| A|)\gamma(v)</math>
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and hence
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:<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>
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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>
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Hence
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:<math>\sum_{v\in V^+}|| A\cap(H+v)|-3^{-d}| A||\ll| A|\eta</math>
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and
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:<math>\sum_{v\in V^+}|| A\cap(H+v)|-3^{-d}| A||^2\ll 3^{-d}| A|^2\eta^2.</math>
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Furthermore, since
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:<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>
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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:
 
To be added:
* Proof
 
 
* Statement of size bound on <math>\Delta</math> from Parseval alone
 
* Statement of size bound on <math>\Delta</math> from Parseval alone
 
* Statement of Chang's theorem
 
* Statement of Chang's theorem
 
* Relation to Lemma 2.8 in Sanders's paper
 
* Relation to Lemma 2.8 in Sanders's paper

Revision as of 03:01, 6 February 2011

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]

Hence

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

and

[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