# A sharp Freiman type estimate for semisums in two and three dimensional Euclidean spaces

created by figalli on 19 Jul 2019

[BibTeX]

Accepted Paper

Inserted: 19 jul 2019
Last Updated: 19 jul 2019

Journal: Ann. Sci. Éc. Norm. Supér.
Year: 2019

Abstract:

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 $\frac12(A+A) - A \ll A$, 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.