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% 31/10/2024 10AM

\begin{fcdefn}[Convolution]
  \glssymboldefn{conv}%
  % Definition 2.11
  Given $f, g : G \to \Cbb$, we define their
  \emph{convolution} $f * g : G \to \Cbb$ by
  \[
    f * g(x) = \EG yG f(y) g(x - y) \qquad
    \forall x \in G
  .\]
\end{fcdefn}

\begin{example}
  % Example 2.12
  Given $A, B \subseteq G$,
  \[
    \indicator{A} \conv \indicator{B}(x)
    = \EG yG \indicator{A}(y) \indicator{B}(x -
    y) = \EG yG \indicator{A}(y) \indicator{x -
    B}(y)
    = \frac{|A \cap (x - B)|}{|G|} = \frac{1}{|G|}
    \rcount_{A + B}(x)
  .\]
  In particular,
  $\operatorname{supp}(\indicator{A} \conv
  \indicator{B}) = A \plus B$.
\end{example}

\begin{fclemma}[]
  \label{lemma_2_13}
  % Lemma 2.13
  Assuming:
  - $f, g : G \to \Cbb$
  Then:
  \[
    \ft{f \conv g}(\gamma) = \ft f(\gamma) \ft
    g(\gamma) \forall \gamma \in \charG
  .\]
\end{fclemma}

\begin{proof}
  \begin{align*}
    \ft{f \conv g}(\gamma)
    &= \EG xG f \conv g(x) \ol{\gamma(x)} \\
    &= \EG xG \EG[y]G f(y) g(\ub{x - y}_{u})
    \ol{\gamma(x)} \\
    &= \EG uG \EG[y]G f(y) g(u) \ol{\gamma(u +
    y)} \\
    &= \ft f(\gamma)\ft g(\gamma) \qedhere
  \end{align*}
\end{proof}

\begin{example}
  % Example 2.14
  \[
    \Ebb_{x + y = z + w} f(x) f(y) \ol{f(z) f(w)}
    = \|\ft f\|_{l^4(\charG)}^4
  .\]
  In particular,
  \[
    \|\ft{\indicator{A}}\|_{l^4(\charG)}^4 =
    \frac{\energy(A)}{|G|^3}
  \]
  for any $A \subseteq G$.
\end{example}

\begin{fcthm}[Bogolyubov's lemma]
  % Theorem 2.15
  Assuming:
  - $A \subseteq \Fp^n$ be a set of density
  $\alpha$
  Then:
  there exists $V \le \Fp^n$ of codimension $\le
  2\alpha^{-2}$ such that $V \subseteq A \plus A
  \minus A \minus A$.
\end{fcthm}

\begin{proof}
  Observe
  \[
    \mdiff 2A - 2A = \supp(\ub{\indicator{A}
    \conv \indicator{A} \conv \indicator{-A} \conv
    \indicator{-A}}_{\eqdef g})
  ,\]
  so wish to find $V \le \Fp^n$ such that $g(x) >
  0$ for all $x \in V$.
  Let $S = \Spec_\rho(\indicator{A})$ with $\rho =
  \sqrt{\frac{\alpha}{2}}$ and let $V = \langle S \rangle^{\perp}$. By
  \cref{lemma_2_10}, $\codim(V) \le |S| \le
  \rho^{-2} \alpha^{-1}$. Fix $x \in V$.
  \begin{align*}
    g(x)
    &= \sum_{t \in \charG[\Fp^n]} \ft g(t) \e(x
    \cdot t / p) \\
    &= \sum_{t \in \charG[\Fp^n]}
    |\ft{\indicator{A}}(t)|^4 \e(x \cdot t / p)
    &&\text{by \cref{lemma_2_13}} \\
    &= \alpha^4 + \sum_{t \neq 0}
    |\ft{\indicator{A}}(t)|^4 \e(x \cdot t / p) \\
    &= \alpha^4
    + \ub{\sum_{t \in S \setminus \{0\}}
    |\ft{\indicator{A}}(t)|^4 \e(x \cdot t /
    p)}_{(1)}
    + \ub{\sum_{t \notin S} |\ft{\indicator{A}}(t)|^4
    \e(x \cdot t / p)}_{(2)}
  \end{align*}
  Note $(1) \ge (\rho \alpha)^4$ since $x \cdot t
  = 0$ for all $t \in S$ and
  \begin{align*}
    |(2)|
    &\le \sup_{t \notin S}
    |\ft{\indicator{A}}(t)|^2 \sum_{t \notin S}
    |\ft{\indicator{A}}|^2 \\
    &\le \sup_{t \in S}
    |\ft{\indicator{A}}(t)|^2 \sum_{t \notin S}
    |\ft{\indicator{A}}|^2 \\
    &\le (\rho \alpha)^2 \|\indicator{A}\|_2^2
    &&\text{by \gls{parse}} \\
    &= \rho^2 \alpha^3
  \end{align*}
  hence $g(x) > 0$ (in fact, $\ge
  \frac{\alpha^4}{2}$) for all $x \in V$ and
  $\codim(V) \le 2\alpha^{-2}$.
\end{proof}

\begin{example}
  % Example 2.16
  The set $A = \{x \in \Fbb_2^n : |x| \ge
  \frac{n}{2} + \frac{\sqrt{n}}{2}\}$ (where $|x|$
  counts the number of 1s in $x$) has density $\ge
  \frac{1}{8}$, but there is no coset $C$ of any
  subspace of codimension $\sqrt{n}$ such that $C
  \subseteq A \plus A (= A \minus A)$.
\end{example}

\begin{fclemma}[]
  \label{lemma_2_17}
  % Lemma 2.17
  Assuming:
  - $A \subseteq \Fp^n$ of density $\alpha$
  - $\rho > 0$
  - $\sup_{t \neq 0} |\ft{\indicator{A}}(t)| \ge
  \rho \alpha$
  Then:
  there exists $V \le \Fp^n$ of codimension $1$
  and $x \in \Fp^n$ such that
  \[
    |A \cap (x + V)| \ge \alpha \left( 1 +
    \frac{\rho}{2} \right) |V|
  .\]
\end{fclemma}

\begin{center}
  \includegraphics[width=0.3\linewidth]{images/e7e79921ff454fb9.png}
\end{center}

\begin{proof}
  Let $t \neq 0$ be such that
  $|\ft{\indicator{A}}(t)| \ge \rho \alpha$, and
  let $V = \langle t \rangle^\perp$. Write $v_j +
  V$ for $j \in [p] =  \{1, 2, \ldots, p\}$ for
  the $p$ distinct cosets $v_j + V = \{x \in \Fp^n
  : x \cdot t = j\}$ of $V$. Then
  \begin{align*}
    \ft{\indicator{A}}(t)
    &= \ft{\balf}(t) \\
    &= \EG x{\Fp^n} (\indicator{A}(x) -
    \alpha)\e(-x \cdot t / p) \\
    &= \EG j{[p]} \EG x{v_j + V} (\indicator{A}(x)
    - \alpha)\e(-j / p) \\
    &= \EG j{[p]} \left(\ub{\frac{|A \cap (v_j + V)|}{|v_j +
    V|} - \alpha}_{= a_j}\right) \e(-j / p)
  \end{align*}
  By triangle inequality, $\EG j{[p]} |a_j| \ge
  \rho \alpha$.
  But note that $\EG j{[p]} a_j = 0$ so $\EG
  j{[p]} a_j + |a_j| \ge \rho \alpha$, hence there
  exists $j \in [p]$ such that $a_j + |a_j| \ge
  \rho \alpha$. Then $a_j \ge \frac{\rho
  \alpha}{2}$.
\end{proof}
