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In particular, if $|A \plus A| \le K|A|$, then
\[
  \energy(A) = \energy(A, A) \ge \frac{|A|^4}{|A
  \plus A|} \ge \frac{|A|^3}{K}
.\]
The converse is \emph{not} true.

\begin{example}
  % Example 1.17
  Let $G$ be your favourite (class of) abelian
  group(s). Then there exist constants $\theta,
  \eta > 0$ such that for all sufficiently large
  $n$, there exists $A \subseteq G$, with $|A| \ge
  n$ satisfying $\energy(A) \ge \eta|A|^3$ and $|A
  \plus A| \ge \theta |A|^2$.
\end{example}

\begin{fcthm}[Balog--Szemeredi--Gowers, Schoen]
  \label{bsg}
  % Theorem 1.18
  Assuming:
  - $A \subseteq G$ is finite
  - $\energy(A) \ge \eta |A|^3$ for some $\eta > 0$
  Then:
  there exists $A' \subseteq A$ of size at least
  $c_1(\eta) |A|$ such that $|A' \plus A'| \le
  \frac{|A'|}{c_2(\eta)}$, where $c_1(\eta)$ and
  $c_2(\eta)$ are polynomial in $\eta$.
\end{fcthm}

\textbf{Idea:} Find $A' \subseteq A$ such that
$\forall a, b \in A'$ such that $a - b$ has many
representations as $(a_1 - a_2) + (a_3 - a_4)$
with $a_i \in A$.

We first prove a technical lemma, using a
technique called ``dependent random choice''.

\begin{fcdefn}[gamma-popular differences]
  \glssymboldefn{Pg}%
  Given $A \subseteq G$ and $\gamma > 0$, let
  \[
    P_\gamma = \{x \in G : |A \cap (x + A)| \ge
    \gamma|A|\}
  \]
  be the set of \emph{$\gamma$-popular differences}
  of $A$.
\end{fcdefn}

\begin{fclemma}[]
  \label{lemma_1_19}
  % Lemma 1.19
  Assuming:
  - $A \subseteq G$ is finite
  - $\energy(A) \ge \eta|A|^3$
  - $c > 0$
  Then:
  there is a subset $X \subseteq A$ of size $|X|
  \ge \eta |A| / 3$ such that for all but a
  $(16c)$-proportion of pairs $(a, b) \in X^2$, $a
  - b \in \Pg[c\eta]$.
\end{fclemma}

\begin{proof}
  Let $U = \{x \in G : |A \cap (x \plus A)| \le
  \half \eta|A|\}$. Then
  \begin{align*}
    \sum_{x \in U} |A \cap (x \plus A)|^2
    &= \half \eta |A| \sum_x |A \cap (x \plus A)|
    \\
    &= \half \eta|A|^3 \\
    &= \half \energy(A)
  \end{align*}
  For $0 \le i \le \left\lceil \log_2 \eta^{-1}
  \right\rceil$, let
  \[
    Q_i = \left\{ x \in G : \frac{|A|}{2^{i + 1}}
    < |A \cap (x \plus A)| \le \frac{|A|}{2^i} \right\}
  ,\]
  and set $\delta_i = \eta^{-1} 2^{-2i}$. Then
  \begin{align*}
    \sum_i \delta_i|Q_i|
    &= \sum_i \frac{|Q_i|}{\eta^{2^{2i}}} \\
    &= \frac{1}{\eta|A|^2} \sum_{i}
    \frac{|A|^2}{2^{2i}} |Q_i| \\
    &= \frac{1}{\eta|A|^2} \sum_{i}
    \frac{|A|^2}{2^{2i}} \sum_{x \notin U}
    \indicator{\left\{ \frac{|A|}{2^{i + 1}} < |A
    \cap (x \plus A)| \le \frac{|A|}{2^i}\right\}}
    \\
    &\ge \frac{1}{\eta|A|^2} \sum_{x \notin U} |A
    \cap (x \plus A)|^2 \\
    &\ge \frac{1}{\eta|A|^2} \cdot \half
    \energy(A)
    &&\left( \sum_{x \in U} |A \cap (x \plus
    A)|^2 \le \half \energy(A) \right)
    \\
    &= \half|A| \tag{$*$} \label{bsg_lemma_eq}
  \end{align*}
  Let $S = \{(a, b) \in A^2 : a - b \notin
  \Pg[c\eta]\}$. Then
  \begin{align*}
    \sum_{i} \sum_{(a, b) \in S} |(A \minus a)
    \cap (A \minus b) \cap Q_i|
    &\le \sum_{(a, b) \in S} \ub{|(A \minus a) \cap (A
    \minus b)|}_{=\ub{|A \cap (a - b + A)| \le
    c\eta|A|}_{\text{by definition of $S$}}} \\
    &\le |S| \cdot c\eta|A| \\
    &\le c\eta |A|^3 \\
    &\le 2c\eta|A|^2 \cdot \half |A| \\
    &\stackrel{\eqref{bsg_lemma_eq}}{\le}
    2c\eta|A|^2 \sum_{i} \delta_i |Q_i|
  \end{align*}
  Hence there exists $i_0$ such that
  \[
    \sum_{(a, b) \in S} |(A \minus a) \cap (A
    \minus b) \cap Q_{i_0}| \le 2 c \eta|A|^2
    \delta_{i_0} |Q_{i_0}|
  .\]
  Let $Q = Q_{i_0}$, $\delta = \delta_{i_0}$,
  $\lambda = 2^{-i_0}$. So
  \[
    \sum_{(a, b) \in S} |(A \minus a) \cap (A
    \minus b) \cap Q| \le 2c\eta \delta|A|^2 |Q|
    \tag{$**$} \label{bsg_lemma_eq_2}
  .\]
  Find $x$ such that $X = |A \cap (A \plus x)|$ is
  large.