  Motivated by the Polynomial Freiman-Ruzsa (PFR) Conjecture, we develop a
theory of locality in sumsets, with applications to John-type approximation and
sets with small doubling. First we show that if $A \subset \mathbb{Z}$ with
$|A+A| \le (1-\epsilon) 2^d |A|$ is non-degenerate then $A$ is covered by
$O(2^d)$ translates of a $d$-dimensional generalised arithmetic progression
($d$-GAP) $P$ with $|P| \le O_{d,\epsilon}(|A|)$; thus we obtain one of the
polynomial bounds required by PFR, under the non-degeneracy assumption that $A$
is not efficiently covered by $O_{d,\epsilon}(1)$ translates of a $(d-1)$-GAP.
We also prove a stability result showing for any $\epsilon,\alpha>0$ that if $A
\subset \mathbb{Z}$ with $|A+A| \le (2-\epsilon)2^d|A|$ is non-degenerate then
some $A' \subset A$ with $|A'|>(1-\alpha)|A|$ is efficiently covered by either
a $(d+1)$-GAP or $O_{\alpha}(1)$ translates of a $d$-GAP. This `dimension-free'
bound for approximate covering makes for a stark contrast with exact covering,
where the required number of translates grows exponentially with $d$. We
further show that if $A \subset \mathbb{Z}$ is non-degenerate with $|A+A| \le
(2^d + \ell)|A|$ and $\ell \le 0.1 \cdot 2^d$ then $A$ is covered by $\ell+1$
translates of a $d$-GAP $P$ with $|P| \le O_d(|A|)$; this is tight, in that
$\ell+1$ cannot be replaced by any smaller number. The above results also hold
for $A \subset \mathbb{R}^d$, replacing GAPs by a suitable common
generalisation of GAPs and convex bodies. In this setting the non-degeneracy
condition holds automatically, so we obtain essentially optimal bounds with no
additional assumption on $A$. These results are all deduced from a unifying
theory, in which we introduce a new intrinsic structural approximation of any
set, which we call the `additive hull', and develop its theory via a refinement
of Freiman's theorem with additional separation properties.
