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 ∣∣ 1 2 (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.