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% 07/11/2024 10AM

\begin{example}[Behrend's example]
  % Example 2.21
  There exists $A \subseteq [N]$ of size at least
  $|A| \ge \exp(-c\sqrt{\log N}) N$ containing no
  non-trivial 3 term arithmetic progressions.
\end{example}

\begin{fclemma}[]
  \label{lemma_2_22}
  % Lemma 2.22
  Assuming:
  - $A \subseteq [N]$ of density $\alpha > 0$
  - $N > 50 \alpha^{-2}$
  - $A$ contains no non-trivial 3 term arithmetic
  progressions
  - $p$ a prime in $\left[ \frac{N}{3},
  \frac{2N}{3} \right]$
  - let $A' = A \cap [p] \subseteq \Zbb / p\Zbb$
  Then:
  one of the following holds:
  \begin{enumerate}[(i)]
    \item $\sup_{t \neq 0}
      |\ft{\indicator{A'}}(t)| \ge
      \frac{\alpha^2}{10}$ (where the Fourier
      coefficient is computed in $\Zbb / p\Zbb$)
    \item There exists an interval $J \subseteq
      [N]$ of length $\ge \frac{N}{3}$ such that
      $|A \cap J| \ge \alpha \left( 1 +
      \frac{\alpha}{400} \right) |J|$
  \end{enumerate}
\end{fclemma}

\begin{proof}
  We may assume that $|A'| = |A \cap [p]| \ge
  \alpha \left( 1 - \frac{\alpha}{200} \right) p$
  since otherwise
  \begin{align*}
    |A \cap [p + 1, N]|
    &\ge \alpha N - \left( \alpha \left( 1 -
    \frac{\alpha}{200} \right) p \right) \\
    &= \alpha (N - p) + \frac{\alpha^2}{200} p \\
    &\ge \left( \alpha + \frac{\alpha^2}{400}
    \right) (N - p)
  \end{align*}
  so we would be in Case (ii) with $J = [p + 1,
  N]$. Let $A'' = A' \cap \left[ \frac{p}{3},
  \frac{2p}{3} \right]$. Note that all 3 term
  arithmetic progressions of the form $(x, x + d,
  x + 2d) \in A' \times A'' \times A''$ are in
  fact arithmetic progressions in $[N]$.

  If $|A' \cap \left[ \frac{p}{3} \right] |$ or
  $|A' \cap \left[ \frac{2p}{3}, p \right]|$ were
  at least $\frac{2}{5} |A'|$, we would again be
  in case (ii). So we may assume that $|A''| \ge
  \frac{|A'|}{5}$.

  Now as in \cref{lemma_2_18} and \cref{thm_2_19},
  \begin{align*}
    \frac{\alpha''}{p}
    &= \frac{|A''|}{p^2} \\
    \T_3(\indicator{A'}, \indicator{A''},
    \indicator{A''}) \\
    &= \alpha' (\alpha'')^2 +
    \sum_{t} \ol{\ft{\indicator{A'}}(t)
    \ft{\indicator{A''}}(t)} \ft{\indicator{2
    \stimes A''}}(t)
  \end{align*}
  where $\alpha' = \frac{|A'|}{p}$ and $\alpha'' =
  \frac{|A''|}{p}$. So as before,
  \[
    \frac{\alpha' \alpha''}{2} \le \sup_{t \neq 0}
    |\indicator{A'}(t)| \cdot \alpha''
  ,\]
  provided that $\frac{\alpha''}{p}  \le \half
  \alpha' (\alpha'')^2$, i.e. $\frac{2}{p} \le
  \alpha' \alpha''$.
  (Check this is satisfied).

  Hence
  \[
    \sup_{t \neq 0} |\ft{\indicator{A'}}(t)| \ge
    \frac{\alpha' \alpha''}{2} \ge \half \left(
    \alpha \left( 1 - \frac{\alpha}{200} \right)
    \right)^2 \cdot \frac{2}{5}
    \ge \frac{\alpha^2}{10}
    . \qedhere
  \]
\end{proof}

\begin{fclemma}[]
  \label{lemma_2_23}
  % Lemma 2.23
  Assuming:
  - $m \in \Nbb$
  - $\varphi : [m] \to \Zbb / p\Zbb$ be given by $x
  \mapsto tx$ for some $t \neq 0$
  - $\eps > 0$
  Then:
  there exists a partition of $[m]$ into
  progressions $P_i$ of length $l_i \in \left[
  \frac{\eps \sqrt{m}}{2}, \eps \sqrt{m} \right]$
  such that
  \[
    \diam(\varphi(P_i)) = \max_{x, y \in P_i}
    |\varphi(x) - \varphi(y)| \le \eps p
  \]
  for all $i$.
\end{fclemma}

\begin{proof}
  Let $u = \left\lfloor \sqrt{m} \right\rfloor$
  and consider $0, t, 2t, \ldots, ut$. By
  Pigeonhole, there exists $0 \le v < w \le u$such
  that $|wt - vt| = |(w - v)t| \le \frac{p}{u}$.
  Set $s = w - v$, so $|st| \le \frac{p}{u}$.
  Divide $[m]$ into residue classes modulo $s$,
  each of which has size at least $\frac{m}{s} \ge
  \frac{m}{4}$. But each residue class can be
  divided into arithmetic progressions of the form
  $a, a + s, \ldots, a + ds$ with $\eps
  \frac{u}{2} < d \le \eps u$. The diameter of the
  image of each progression under $\varphi$ is
  $|dst| \le d \frac{p}{u} \le \eps u \frac{p}{u}
  = \eps p$.
\end{proof}

\begin{fclemma}[]
  % Lemma 2.24
  Assuming:
  - $A \subseteq [N]$ of density $\alpha > 0$
  - $p$ a prime in $\left[ \frac{N}{3},
  \frac{2N}{3} \right]$
  - let $A' = A \cap [p] \subseteq \Zbb / p\Zbb$
  - $|\ft{\indicator{A'}}(t)| \ge
  \frac{\alpha^2}{20}$ for some $t \neq 0$
  Then:
  there exists a progression $P \subseteq [N]$ of
  length at least $\alpha^2 \frac{\sqrt{N}}{500}$
  such that $|A \cap P| \ge \alpha \left( 1 +
  \frac{\alpha}{80} \right) |P|$.
\end{fclemma}