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\newpage
\section{A proof of Marton's conjecture in
\texorpdfstring{$\Fbb_2^n$}{F\_2\^{}n}}

We shall prove the following theorem.

\begin{fcthm}[Green, Manners, Tao, Gowers]
  \label{thm:7.1}
  There is a polynomial $p$ with the following
  property:

  \divcloze{If $n \in \Nbb$ and $A \subset \Fbb_2^n$ is such
  that $|A + A| \le C|A|$, then there is a
  subspace $H \subset \Fbb_2^n$ of size at most
  $|A|$ such that $A$ is contained in the union of
  at most $p(C)$ translates of $H$. (Equivalently,
  there exists $K \subset \Fbb_2^n$, $|K| \le
  p(C)$ such that $A \subset K + H$).}
\end{fcthm}

This is known as ``Polynomial Freiman--Ruzsa''.

In fact, we shall prove the following statement.

\begin{fcthm}[Entropic Polynomial Freiman--Ruzsa]
  \label{thm:7.2}
  There exists an absolute constant $\alpha$
  satisfying the following:

  Let $G = \Fbb_2^n$ and let $X, Y$ be $G$-valued
  random variables. Then there exists a subsgroup
  $H$ of $G$ such that
  \[
    \entd{X}{U_H} + \entd{U_H}{Y}
    \le \alpha \entd XY
  \]
  where $U_H$ is the uniform distribution on $H$.
\end{fcthm}

\begin{fclemma}[]
  \label{lemma:7.3}
  Assuming:
  - $X$ a discrete random variable (and write
  $p_x$ for $\Pbb(X = x)$)
  Then:
  there exists $x$ such that $p_x \ge 2^{-\ent
  X}$.
\end{fclemma}

\begin{proof}
  If not, then
  \[
    \ent X
    = \sum_x p_x \log \left( \frac{1}{p_x} \right)
    > \ent X \sum_x p_x
    = \ent X
  ,\]
  contradiction.
\end{proof}

\begin{fcprop}[]
  \label{prop:7.4}
  \cref{thm:7.2} implies \cref{thm:7.1}.
\end{fcprop}

\begin{proof}
  Let $A \subset \Fbb_2^n$, $|A + A| \le C|A|$.
  Let $X$ and $Y$ be independent copies of $U_A$.
  Then by \cref{thm:7.2}, there exists $H$ (a
  subgroup) such that
  \[
    \entd{X}{U_H} + \entd{U_H}{X}
    \le \alpha \entd XY
  \]
  so
  \[
    \entd{X}{U_H}
    \le \frac{\alpha}{2} \entd XY
  .\]
  But
  \begin{align*}
    \entd XY
    &= \ent{U_A - U_A'} - \ent{U_A} \\
    &= \ent{U_A + U_A'} - \ent{U_A}
    &&\text{(characteristic 2)} \\
    &\le \log (C|A|) - \log |A| \\
    &= \log C
  \end{align*}
  So $\entd{X}{U_H} \le \frac{\alpha \log C}{2}$.
  Therefore
  \begin{align*}
    \ent{X + U_H}
    &\le \half \ent X + \half \ent{U_H} +
    \frac{\alpha \log C}{2} \\
    &= \half \log |A| + \half \log |H| +
    \frac{\alpha \log C}{2}
  \end{align*}
  Therefore, by \cref{lemma:7.3}, there exists $z$
  such that
  \[
    \Pbb(X + U_H = z)
    \ge |A|^{-\half} |H|^{-\half}
    C^{-\frac{\alpha}{2}}
  .\]
  But
  \[
    \Pbb(X + U_H = z)
    = \frac{|A \cap (z - H)|}{|A| |H|} = \frac{|A
    \cap (z + H)|}{|A| |H|}
  \]
  (using characteristic 2). So there exists $z \in
  G$ such that
  \[
    |A \cap (z + H)| \ge C^{-\frac{\alpha}{2}}
    |A|^{\half} |H|^{\half}
  .\]
  Let $B = A \cap (z + H)$. By the
  \nameref{lemma:6.2}, we can cover $A$ by at most
  $\frac{|A + B|}{|B|}$ translates of $B + B$. But
  $B \subset z + H$ so $B + B \subset H + H = H$,
  so $A$ can be covered by at most $\frac{|A +
  B|}{|B|}$ translates of $H$.

  But using $B \subset A$,
  \[
    |A + B|
    \le |A + A|
    \le C|A|
  .\]
  So
  \[
    \frac{|A + B|}{|B|}
    \le \frac{C|A|}{C^{-\frac{\alpha}{2}}
    |A|^{\half} |H|^{\half}}
    = C^{\frac{\alpha}{2} + 1}
    \frac{|A|^{\half}}{|H|^{\half}}
  .\]
  Since $B$ is contained in $z + H$,
  \[
    |H|
    \ge C^{-\frac{\alpha}{2}} |A|^{\half}
    |H|^{\half}
  \]
  so $|H| \ge C^{-\alpha} |A|$, so
  \[
    C^{\frac{\alpha}{2} + 1}
    \frac{|A|^{\half}}{|H|^{\half}}
    \le C^{\alpha + 1}
  .\]
  If $|H| \le |A|$ then we are done. Otherwise,
  since $B \subset A$,
  \[
    |A|
    \ge C^{-\frac{\alpha}{2}} |A|^{\half}
    |H|^{\half}
  \]
  so $|H| \le C^{\alpha} |A|$.

  Pick a subgroup $H'$ of $H$ of size between
  $\frac{|A|}{2}$ and $|A|$. Then $H$ is a union
  of at most $2C^{\alpha}$ translates of $H'$, so
  $A$ is a union of at most $2C^{2\alpha + 1}$
  translates of $H'$.
\end{proof}

Now we reduce further. We shall prove the
following statement:

\begin{fcthm}[EPFR$'$]
  \label{thm:7.5}
  % \label{thm:7.3}
  There is a constant $\eta > 0$ such that \cloze{if $X$
  and $Y$ are any two $\Fbb_2^n$-valued random
  variables with $\entd XY > 0$, then there exists
  $\Fbb_2^n$-valued random variables $U$ and $V$
  such that}
  \divcloze{
    \[
      \entd UV + \eta(\entd UX + \entd VY)
      < \entd XY
    .\]
  }
\end{fcthm}
