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\begin{fcthm}[Szemeredi's Theorem for 4-APs]
  \label{thm_4_12}
  % [Szemer\'edi's Theorem for 4-APs]
  % Theorem 4.12
  Assuming:
  - $A \subseteq \Fbb_5^n$ a set containing
  no non-trivial 4 term arithmetic progressions
  Then:
  $|A| = o(5^n)$.
\end{fcthm}

\textbf{Idea:} By \cref{prop_4_11} with $f = \balf =
\indicator{A} - \alpha$,
\[
  \T_4(
    \ub{\indicator{A}}_{\balf + \alpha},
    \ub{\indicator{A}}_{\balf + \alpha},
    \ub{\indicator{A}}_{\balf + \alpha},
    \ub{\indicator{A}}_{\balf + \alpha}
  )
  - \alpha^4
  = \T_4(\balf, \balf, \balf, \balf)
  + \cdots
\]
where $\cdots$ consists of $14$ other terms in
which between one and three of the inputs are
equal to $\balf$.

These are controlled by
\[
  \|\balf\|_{U^2(\Fbb_5^n)}
  \le \unorm{\balf}
,\]
whence
\[
  |\T_4(\indicator{A}, \indicator{A}, \indicator{A},
  \indicator{A}) - \alpha^4|
  \le 15 \unorm{\balf}
.\]
So if $A$ contains no non-trivial 4 term
arithmetic progressions and $5^n > 2\alpha^{-3}$,
then $\unorm{\balf} \ge \frac{\alpha^4}{30}$.

What can we say about functions with large $U^3$
norm?

\begin{example}
  \label{eg_4_13}
  % Example 4.13
  Let $M$ be an $n \times n$ symmetric matrix with
  entries in $\Fbb_5$. Then $f(x) = e(x^\top M x /
  5)$ satisfies $\unorm{f} = 1$.
\end{example}

\begin{fcthm}[$U^3$ inverse theorem]
  \label{thm_4_14}
  % Theorem 4.14
  Assuming:
  - $f : \Fbb_5^n \to \Cbb$
  - $\|f\|_{L^\infty(\Fbb_5^n)} \le 1$
  - $\unorm{f} \ge \delta$ for some $\delta > 0$
  Then:
  there exists a symmetric $n \times n$ matrix $M$
  with entries in $\Fbb_5$ and $b \in \Fbb_5^n$
  such that
  \[
    |\Ebb_x f(x) e((x^\top M x + b^\top x) / p)|
    \ge c(\delta)
  \]
  where $c(\delta)$ is a polynomial in $\delta$.

  In other words, $|\langle f, \phi\rangle| \ge
  c(\delta)$ for $\phi(x) = e((x^\top M x + b^\top
  x) / p)$ and we say ``$f$ correlates with a
  quadratic phase function''.
\end{fcthm}

\begin{proof}[Proof (sketch)]
  Let $\Delta_h f(x)$ denote $f(x) \ol{f(x + h)}$.

  $\unorm{f} = (\Ebb_h \|\Delta_h
  f\|_{U^2}^4)^{\frac{1}{8}}$.

  \begin{enumerate}[STEP 1:]
    \item Weak linearity. See reference.
    \item Strong linearity. We will spend the rest
      of the lecture discussing this in detail.
    \item Symmetry argument. Problem 8 on Sheet 3.
    \item Integration step. Problem 9 on Sheet 3.
  \end{enumerate}
  STEP 1: If $\unorm{f}^8 = \Ebb_h
  \|\Delta_h\|_{U^2}^4 \ge \delta^8$, then for at
  least a $\frac{\delta^8}{2}$-proportion of $h
  \in \Fbb_5^n$, $\frac{\delta^8}{2} \le
  \|\Delta_h f\|_{U^2}^4 \le \|\ft{\Delta_h
  f}\|_{l^\infty}^2$. So for each such $h \in
  \Fbb_5^n$, there exists $t_h$ such that
  $|\ft{\Delta_h f}(t_h)|^2 \ge
  \frac{\delta^8}{2}$.

  \begin{fcprop}[]
    \label{prop_4_15}
    % Proposition 4.15
    Assuming:
    - $f : \Fbb_5^n \to \Cbb$
    - $\|f\|_\infty \le 1$
    - $\unorm{f} \ge \delta$
    - $|\Fbb_5^n| = \Omega_\delta(1)$
    Then:
    there exists $S \subseteq \Fbb_5^n$ with $|S|
    = \Omega_\delta(|\Fbb_5^n|)$ and a function
    $\phi : S \to \charG[\Fbb_5^n]$ such that
    \begin{enumerate}[(i)]
      \item $|\ft{\Delta_h f}(\phi(h))| =
        \Omega_\delta(1)$;
      \item There are at least
        $\Omega_\delta(|\Fbb_5^n|^3)$ quadruples
        $(s_1, s_2, s_3, s_4) \in S^4$ such that
        $s_1 + s_2 = s_3 + s_4$ and $\phi(s_1) +
        \phi(s_2) + \phi(s_4)$.
    \end{enumerate}
  \end{fcprop}
  STEP 2: If $S$ and $\phi$ are as above, then
  there is a linear function $\psi : \Fbb_5^n \to
  \charG[\Fbb_5^n]$ which coincides with $\phi$
  for many elements of $S$.

  \begin{fcprop}[]
    \label{prop_4_16}
    % Proposition 4.16
    Assuming:
    - $S$ and $\phi$ given as in \cref{prop_4_15}
    Then:
    there exists $n \times n$ matrix $M$ with
    entries in $\Fbb_5$ and $b \in \Fbb_5^n$ such
    that $\psi(x) = Mx + b$ ($\psi : \Fbb_5^n \to
    \charG[\Fbb_5^n]$) satisfies $\psi(x) =
    \phi(x)$ for $\Omega_\delta(|\Fbb_5^n|)$
    elements $x \in S$.
  \end{fcprop}

  \begin{proof}
    Consider the graph of $\phi$, $\Gamma = \{(h,
    \phi(h)) : h \in S\} \subseteq \Fbb_5^n \times
    \charG[\Fbb_5^n]$. By \cref{prop_4_15},
    $\Gamma$ has $\Omega_\delta(|\Fbb_5^n|^3)$
    additive quadruples.

    By \nameref{bsg}, there exists $\Gamma'
    \subseteq \Gamma$ with $|\Gamma'| =
    \Omega_\delta(|\Gamma|) =
    \Omega_\delta(|\Fbb_5^n|)$ and $|\Gamma' +
    \Gamma'| = O_\delta(|\Gamma'|)$. udefine $S'
    \subseteq S$ by $\Gamma' = \{(h, \phi(h)) : h
    \in S'\}$ and note $|S'| =
    \Omega_\delta(|\Fbb_5^n|)$.

    By \nameref{thm_1_11} applied to $\Gamma'
    \subseteq \Fbb_5^n \times \charG[\Fbb_5^n]$,
    there exists a subspace $H \le \Fbb_5^n \times
    \charG[\Fbb_5^n]$ with $|H| =
    O_\delta(|\Gamma'|) = O_\delta(|\Fbb_5^n|)$
    such that $\Gamma' \subseteq H$.

    Denote by $\pi : \Fbb_5^n \times
    \charG[\Fbb_5^n] \to \Fbb_5^n$ the projection
    onto the first $n$ coordinates. By
    construction, $\pi(H) \supseteq S'$. Moreover,
    since $|S'| = \Omega_\delta(|\Fbb_5^n|)$,
    \[
      |\ker(\pi|_H)| = \frac{|H|}{|\Im(\pi|_H)|}
      = \frac{O_\delta(|\Fbb_5^n|)}{|S'|}
      = O_\delta(1)
    .\]
    We may thus partition $H$ into $O_\delta(1)$
    cosets of some subspace $H^*$ such that
    $\pi|_H$ is injective on each coset. By
    averaging, there exists a coset $x + H^*$ such
    that
    \[
      |\Gamma' \cap (x + H^*)| =
      \Omega_\delta(|\Gamma'|)
      = \Omega_\delta(|\Fbb_5^n|)
    .\]
    Set $\Gamma'' = \Gamma' \cap (x + H^*)$, and
    define $S''$ accordingly.

    Now $\pi|_{x + ^*}$ is injective and
    surjective onto $V \defeq \Im(\pi|_{x +
    H^*})$. This means there is an affine linear
    map $\psi : V \to \charG[\Fbb_5^n]$ such that
    $(h, \psi(h)) \in \Gamma''$ for all $h \in
    S''$.
  \end{proof}
  Then do steps 3 and 4.
\end{proof}

\begin{flashcard}
  What to do if you have lots of additive
  quadruples?

  \cloze{
    Balog-Szemeredi-Gowers
  }
\end{flashcard}

\begin{flashcard}
  What to do if you have small doubling constant?

  \cloze{
    Freiman-Ruzsa (or Polynomial-Freiman-Ruzsa)
  }
\end{flashcard}