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

  • Authors:

  • Yonutz V. Stanchescu

  • Affiliations:

  • School of Mathematical Sciences, Tel Aviv University 69978 Ramat Aviv, Tel Aviv, Israel

  • Venue:
  • Discrete Mathematics
  • Year:
  • 2002

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Abstract

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 ≤ 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.