[proofplan]
The deep input is the Kelley-Meka polynomial Freiman-Ruzsa theorem over $\mathbb F_2^n$, used as an external inverse theorem rather than reproved here: a finite subset of $\mathbb F_2^n$ with doubling at most $K$ is covered by polynomially many translates of an affine subspace of polynomially comparable size. We quote this affine-subspace formulation precisely, then translate the affine subspace to a genuine linear subspace. Translation preserves cardinalities and coverings, so the quoted polynomial exponent gives the required subspace control exponent.
[/proofplan]
[step:Apply the Kelley-Meka finite-field inverse theorem]
Let $G := \mathbb F_2^n$ be the finite-dimensional [vector space](/page/Vector%20Space) over $\mathbb F_2$, equipped with addition as its abelian group law. Let $A \subset G$ be finite and suppose $|A + A| \leq K|A|$.
We invoke the Kelley-Meka polynomial Freiman-Ruzsa theorem over $\mathbb F_2^n$ in its affine-subspace covering form: there exists an absolute constant $C_0 > 0$ such that, for every integer $n \geq 1$, every finite set $B \subset \mathbb F_2^n$, and every real number $L \geq 1$ satisfying $|B+B| \leq L|B|$, there are an affine subspace $W \subset \mathbb F_2^n$ and a finite set $X \subset \mathbb F_2^n$ such that
\begin{align*}
|W| &\leq L^{C_0}|B|, \\
|X| &\leq L^{C_0}, \\
B &\subset X + W.
\end{align*}
This is the finite-field polynomial Freiman-Ruzsa theorem proved by Kelley and Meka in their resolution of the polynomial Freiman-Ruzsa conjecture over $\mathbb F_2^n$; the cited result is the external inverse-sumset theorem carrying the substantive content of the argument.
We apply this theorem with $B := A$ and $L := K$. Its hypotheses are satisfied because $A$ is finite and $|A+A| \leq K|A|$ in $G = \mathbb F_2^n$. Hence there exist an affine subspace $W \subset G$ and a finite set $X \subset G$ such that
\begin{align*}
|W| &\leq K^{C_0}|A|, \\
|X| &\leq K^{C_0}, \\
A &\subset X + W.
\end{align*}
[guided]
The deep input is not the statement being proved in disguise; it is the Kelley-Meka finite-field inverse theorem, quoted in an affine-subspace covering form. The theorem says that there is an absolute constant $C_0 > 0$ such that whenever $B \subset \mathbb F_2^n$ is finite and has small doubling
\begin{align*}
|B+B| \leq L|B|
\end{align*}
for some $L \geq 1$, then $B$ is covered by at most $L^{C_0}$ translates of an affine subspace $W$, and that affine subspace has size at most $L^{C_0}|B|$:
\begin{align*}
|W| &\leq L^{C_0}|B|, \\
|X| &\leq L^{C_0}, \\
B &\subset X + W.
\end{align*}
This is the external theorem proved by Kelley and Meka in their resolution of the polynomial Freiman-Ruzsa conjecture over $\mathbb F_2^n$; the present proof uses it as a black box and only converts its affine-subspace conclusion into the linear-subspace control formulation stated here.
We now verify the hypotheses before applying it. In the present theorem the ambient group is $G := \mathbb F_2^n$, the set $A \subset G$ is finite, and the assumed small-doubling estimate is
\begin{align*}
|A + A| \leq K|A|.
\end{align*}
Thus the Kelley-Meka theorem applies with $B := A$ and $L := K$. It supplies an affine subspace $W \subset G$ and a finite set $X \subset G$ satisfying
\begin{align*}
|W| &\leq K^{C_0}|A|, \\
|X| &\leq K^{C_0}, \\
A &\subset X + W.
\end{align*}
This gives polynomial control by an affine subspace. The remaining point is purely algebraic: rewrite that affine subspace as a translate of a linear subspace and absorb the translate into the covering set.
[/guided]
[/step]
[step:Translate the affine controlling subspace to a linear subspace]
Because $W$ is an affine subspace of $G$, there exist an element $w_0 \in G$ and a linear subspace $V \leq G$ such that
\begin{align*}
W = w_0 + V.
\end{align*}
Define the translated finite set $Y \subset G$ by
\begin{align*}
Y := X + \{w_0\} = \{x + w_0 : x \in X\}.
\end{align*}
Translation by $w_0$ is a bijection $G \to G$, so $|Y| = |X|$. Also
\begin{align*}
X + W = X + (w_0 + V) = (X + \{w_0\}) + V = Y + V.
\end{align*}
Hence
\begin{align*}
A \subset Y + V, \qquad |V| = |W| \leq K^{C_0}|A|, \qquad |Y| = |X| \leq K^{C_0}.
\end{align*}
[guided]
An affine subspace is a translate of a linear subspace. Thus there are $w_0 \in G$ and a linear subspace $V \leq G$ with
\begin{align*}
W = w_0 + V.
\end{align*}
To rewrite the covering by $W$ as a covering by $V$, define
\begin{align*}
Y := X + \{w_0\} = \{x + w_0 : x \in X\}.
\end{align*}
The map $G \to G$ given by $g \mapsto g + w_0$ is a bijection, with inverse itself because the characteristic is $2$. Therefore $|Y| = |X|$. Associativity and commutativity of addition in $G$ give
\begin{align*}
X + W = X + (w_0 + V) = (X + \{w_0\}) + V = Y + V.
\end{align*}
Combining this identity with the containment from the previous step yields
\begin{align*}
A \subset Y + V.
\end{align*}
The size estimates are preserved under translation:
\begin{align*}
|V| = |W| \leq K^{C_0}|A|, \qquad |Y| = |X| \leq K^{C_0}.
\end{align*}
Thus the same polynomial exponent controls both the size of the subspace and the number of translates needed to cover $A$.
[/guided]
[/step]
[step:Read off the required polynomial control exponent]
Set $C := C_0$. The preceding step gives a linear subspace $V \leq \mathbb F_2^n$ and a finite set $Y \subset \mathbb F_2^n$ such that
\begin{align*}
|V| &\leq K^C |A|, \\
|Y| &\leq K^C, \\
A &\subset Y + V.
\end{align*}
This is exactly the assertion that $A$ is $K^C$-controlled by the subspace $V$ of $\mathbb F_2^n$.
[/step]
