# 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 $A$ 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. We shall say later on why this is significant.

## The nd-estimate

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 $V \leq \mathbb{F}_3^n$ we write

$V^{\perp} = \{ \gamma \in \widehat{\mathbb{F}_3^n} : \gamma(x) = 1 \ \forall x \in V \}$

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

Proposition 1 Let $A \subset \mathbb{F}_3^n$ be a set with density $\alpha$, and let $0 \leq \delta, \eta \leq 1$ be parameters. Set
$\Delta = \{ \gamma \in \widehat{G} : | \widehat{1_A}(\gamma) | \geq \delta \alpha \} \setminus \{ 0_{\widehat{\mathbb{F}_3^n}} \}$.
Let $V \leq \mathbb{F}_3^n$ be a subspace. Then
• either $A$ has density at least $\alpha(1 + \eta)$ on a translate of $V$,
• or $|\Delta \cap V^{\perp}| \leq 3\eta \delta^{-2}$; in fact $\sum_{\gamma \in V^{\perp}} |\widehat{(1_A - \alpha)}(\gamma)|^2 \leq 3\eta \alpha^2$.

In other words, if the large spectrum of a set is somewhat concentrated in some subspace then one can find a density increment on a translate of the annihilator of that subspace.

## Proof

Let us write $\mu_V = \frac{|\mathbb{F}_3^n|}{|V|}1_V$ for the scaled indicator function of $V$ normalized so that $\mathbb{E}_x \mu_V(x) = 1$. If

$1_A*\mu_V(x) \gt \alpha(1 + \eta)$

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

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

By Parseval's identity, this equals

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

which proves the result.

## Comparison with other results about the large spectrum of a set

The main ingredient in deriving the nd-estimate is Parseval's identity. This identity also has the following useful consequence: letting $\Delta$ be as above, we have

$|\Delta| \delta^2 \alpha^2 \leq \sum_{\gamma \in \widehat{\mathbb{F}_3^n}} |\widehat{1_A}(\gamma)|^2 = \mathbb{E}_x 1_A(x)^2 = \alpha$,

whence

$|\Delta| \leq \alpha^{-1} \delta^{-2}$,

which should be compared to the bound on $| \Delta \cap V^{\perp} |$ given by the nd-estimate.

There is another useful result about the large spectrum of a set known as Chang's theorem. Informally, this says that the largest size of a linearly independent set in the large spectrum $\Delta$ cannot be too large; formally, the largest independent set has size at most $C\log(\alpha^{-1}) \delta^{-2}$. Unfortunately this statement becomes trivial with the parameters needed for the Bateman-Katz argument ($\delta \sim \alpha \sim 1/n^{1+\epsilon}$). Nevertheless, there is a generalization of Chang's theorem due to Shkredov that gives a lower bound for the number of additive $(2m)$-tuples in the large spectrum of a set, which is used in Section 4 of the Bateman-Katz paper.

By contrast, the nd-estimate is something like a statement in the opposite direction: it says that there are quite a lot of linearly independent characters in $\Delta$, or else there is a density increment. Specifically, if we have picked $\gamma_1, \ldots, \gamma_d$ from $\Delta$, then

$| \Delta \cap \langle \gamma_1, \ldots, \gamma_d \rangle | \leq 3\eta \delta^{-2}$

unless we get a density increment on a (particular) subspace of codimension at most $d$. For suitable parameter choices, this says that there are a lot of characters in the large spectrum that are linearly independent of $\gamma_1, \ldots, \gamma_d$, which is very important in Section 5 of the paper. (Note: for this type of conclusion to hold we need to know that the large spectrum $\Delta$ is quite large, which follows from the assumption that we may make in the $\mathbb{F}_3^n$ context that $A$ has no particularly large non-trivial Fourier coefficients.)