Planar sets containing no three collinear points and non-averaging sets of integers Original Research Article

Let A⊆Z2 be a finite set of lattice points and let |A|=n. We prove that if A does not contain any three collinear points, then |A±A|⪢n(log n)δ. Here δ can be every positive absolute constant δ<18. This lower bound provides an answer to an old question of Freiman. Some further related questions on non-averaging sets of integers are posed and discussed.