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

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

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

and hence

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

If we let $V^+$ be the subset of $V$ for which each of the squared summands is positive, then either $A$ has the required density increment on a translate of $H$ (which has codimension $d$), or

$|| A\cap(H+v)|-3^{-d}| A||\ll 3^{-d}| A|\eta.$

Hence

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

and

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

Furthermore, since

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

defining $V^-$ similarly and combining the trivial estimate

$|| A\cap(H+v)|-3^{-d}| A||\leq3^{-d}| A|$

for $v\in V^-$ with the above gives

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

Combining these sum estimates gives

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

and hence

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

Recalling the definition of $\Delta$, we have

$|\Delta\cap W|\delta^2\alpha^2\ll\sum_{\gamma\in\Delta\cap W}|\widehat{1_A}(\gamma)|^2\ll\alpha^2\eta.$

• Statement of size bound on $\Delta$ from Parseval alone
• Statement of Chang's theorem
• Relation to Lemma 2.8 in Sanders's paper