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

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⊂R and ∣∣12(A+A)∣∣−|A|≪|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. © 2021 Société Mathématique de France