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

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Let A⊆ Z 2 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< 1 8 . 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.