Locality in Sumsets

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.