Inserted: 19 jul 2019
Last Updated: 7 oct 2022
Journal: Ann. Sci. Éc. Norm. Supér.
Freiman's Theorem is a classical result in additive combinatorics
concerning the approximate structure of sets of integers that contain
a high proportion of their internal sums. As a consequence, one can
deduce an estimate for sets of real numbers: ''If $A\subset \mathbb R$ and $
$, then $A$ is close to its convex hull.'' In this paper we prove a sharp form of the analogous result in dimensions 2 and 3.