= Solution
The <Dense Bogolyubov-Ruzsa lemma> says that, for every $\alpha>0$, if $S$ has density at least $\alpha$ in a cyclic group of prime order, then $2S-2S$ contains a proper <generalized arithmetic progression> of rank $O_\alpha(1)$ and size $\Omega_\alpha(|G|)$.
Now suppose $A\subseteq\mathbb Z$ and $|A+A|\leq K|A|$. The <Ruzsa modelling lemma>, taken at a sufficiently high fixed Freiman order, supplies $A'\subseteq A$ with $|A'|\geq|A|/2$ and a Freiman isomorphism from $A'$ to a subset $S\subseteq\mathbb Z/q\mathbb Z$, where $q$ is prime, $q=O_K(|A|)$, and $|S|/q\gg_K1$. Apply the dense Bogolyubov-Ruzsa lemma to $S$ and transfer the resulting progression back through the Freiman model. We obtain a proper progression
$$
P_0\subseteq2A'-2A'\subseteq2A-2A
$$
of rank $O_K(1)$ and size $|P_0|\gg_K|A|$.
The <Plünnecke-Ruzsa inequality> gives
$$
|A+P_0|\leq|3A-2A|\leq K^5|A|.
$$
There are $|A||P_0|$ pairs $(a,p)$ with $a\in A$ and $p\in P_0$, distributed among the sums in $A+P_0$. Some $x$ therefore has at least $K^{-5}|P_0|\gg_K|A|$ representations $x=a+p$. Equivalently,
$$
|A\cap(x-P_0)|\gg_K|A|.
$$
Set $P=x-P_0$. Translation and negation preserve properness and rank. Moreover $P_0\subseteq2A-2A$ and another use of Plünnecke gives $|P|\leq K^4|A|$. Thus
$$
|A\cap P|\gg_K|A|\gg_K|P|,
$$
as required.
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