%! TEX root = AC.tex
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% 21/11/2024 10AM

  So we have
  \[
    \Ebb_{(z_1, \ldots, z_m) \in A^m} \EG yG
    \left| \sum_{i = 1}^{m} Z_i(y) \right|^p
    = O \left( p^{p / 2} m^{p / 2 - 1} \sum_{i =
    1}^{m} (2 \|f\|_{L^p(G)})^p \right)
    = O((4p)^{p / 2} m^{p / 2} \|f\|_{L^p(G)}^p)
  .\]
  Choose $m = O(\eps^{-2} p)$ so that the RHS is
  at most $(\frac{\eps}{4} \|f\|_{L^p(G)})^p$.
  whence
  \[
    \Ebb_{(z_1, \ldots, z_m) \in A^m} \ub{\EG yG
    \left| \frac{1}{m} \sum_{i = 1}^{m} \tau_{-zi}
    f(y) - f \conv \cmeas(y)\right|^p}_{= (*)}
    = O((4p)^{p / 2} m^{p / 2} \|f\|_{L^p(G)}^p)
    = \left( \frac{\eps}{4} \|f\|_{L^p(G)}
    \right)^p
  .\]
  Write
  \[
    L = \left\{z = (z_1, \ldots, z_m) \in A^m : (*) \le
    \left( \frac{\eps}{2} \|f\|_{L^p(G)} \right)^p\right\}
  .\]
  By Markov inequality, since
  \[
    \Ebb (*) \le \left( \frac{\eps}{4}
    \|f\|_{L^p(G)} \right)^p
    = 2^{-p} \left( \frac{\eps}{2}\|f\|_{L^p(G)}
    \right)^p
  ,\]
  we have
  \[
    \frac{|A^m \setminus L|}{|A^m|}
    = \Pbb\left((*) \ge \left(
    \frac{\eps}{2}\|f\|_{L^p(G)} \right)^p \right)
    \le \Pbb((*) \ge 2^p \Ebb(*))
    \le 2^{-p}
  \]
  so $|L| \ge \left( 1 - \frac{1}{2^p} \right)
  |A|^m \ge \half |A|^m$. Let
  \[
    D = \{\ub{(b, b, \ldots, b)}_{m} : b \in B\}
  .\]
  Now $L \plus D \subseteq (A \plus B)^m$, whence
  \[
    |L \plus D|
    \le |A \plus B|^m
    \le K^m |A|^m
    \le 2K^m |L|
  .\]
  By \cref{lemma_1_16},
  \[
    \energy(L, D)
    \ge \frac{|L|^2 |D|^2}{|L \plus D|}
    \ge
    \half K^{-m} |D|^2 |L|
  \]
  so there are at least $\frac{|D|^2}{2K^m}$ pairs
  $(d_1, d_2) \in D \times D$ such that $\rcount_{L
  \minus L}(d_2 - d_1) > 0$. In particular, there
  exists $b \in ub$ and $X \subseteq B - b$ of
  size $|X| \ge \frac{|D|}{2K^m} =
  \frac{|B|}{2K^m}$ such that for all $x \in X$,
  there exists $l_2(x) \in L$ such that for all $i
  \in [m]$, $l_1(x)_i - l_2(x)_i = x$. But then
  for each $x \in X$, by the triangle inequality,
  \begin{align*}
    \|\tau_{-x} f \conv \cmeas - f \conv
    \cmeas\|_{L^p(G)}
    &\le \left\| \tau_{-x} f \conv \cmeas -
    \tau_{-x} \left( \frac{1}{m} \sum_{i = 1}^{m}
    \tau_{-l_2(x)_i} f \right) \right\|_{L^p(G)}
    \\
    &~~+ \left\| \tau_{-x} \left( \frac{1}{m} \sum_{i =
    1}^m \tau_{-l_2(x_i)} f \right) - f \conv
    \cmeas \right\|_{L^p(G)} \\
    &= \left\| f \conv \cmeas - \frac{1}{m} \sum_{i
    = 1}^{m} \tau_{-l_2(x)_i} f \right\|_{L^p(G)}
    \\
    &~~+ \left\| \frac{1}{m} \sum_{i = 1}^{m}
    \tau_{-x - l_2(x)_i} f - f \conv \cmeas
    \right\|_{L^p(G)} \\
    &\le 2 \cdot \frac{\eps}{2} \|f\|_{L^p(G)}
  \end{align*}
  by definion of $L$.
\end{proof}

\begin{fcthm}[Bogolyubov again, after Sanders]
  \label{thm_3_10}
  % Theorem 3.10
  Assuming:
  - $A \subseteq \Fbb_p^n$ of density $\alpha > 0$
  Then:
  there exists a subspace $V \le \Fbb_p^n$ of
  codimension $O(\log^4 \alpha^{-1})$ such tht $V
  \subseteq A \plus A \minus A \minus A$.
\end{fcthm}

Almost periodicity is also a key ingredient in
recent work of Kelley and Meka, showing that any
$A \subseteq [N]$ containing no non-trivial 3 term
arithmetic progressions has size $|A| \le \exp(-C
\log^{\frac{1}{11}} N)N$.

\newpage
\section{Further Topics}

In $\Fbb_p^n$, we can do much better.

\begin{fcthm}[Ellenberg-Gijswijt, following
  Croot-Lev-Pach]
  \label{thm_4_1}
  % Theorem 4.1
  Assuming:
  - $A \subseteq \Fbb_3^n$ contains no
  non-trivial 3 term arithmetic progressions
  Then:
  $|A| = o(2.756)^n$.
\end{fcthm}

\begin{notation*}
  \glssymboldefn{M}%
  Let $M_n$ be the set of monomials in $x_1,
  \ldots, x_2$ whose degree in each variable is at
  most $2$. Let $V_n$ be the vector space over
  $\Fbb_3$ whose basis is $M_n$. For any $d \in
  [0, 2n]$, write $M_n^d$ for the set of monomials
  in $M_n$ of (total) degree at most $d$, and
  $V_n^d$ for the corresponding vector space. Set
  $m_d = \dim(V_n^d) = |M_n^d|$.
\end{notation*}

\begin{fclemma}[]
  \label{lemma_4_2}
  % Lemma 4.2
  Assuming:
  - $A \subseteq \Fbb_3^n$
  - $P \in \V_n^d$ is a polynomial
  - $P(a + a') = 0$ for all $a \neq a' \in A$
  Then:
  \[
    |\{a \in A : P(2a) \neq 0\}| \le 2 \m_{d / 2}
  .\]
\end{fclemma}