Sum of sets in several dimensions


LetA, B be finite sets in ℝd with|A|=m≤|B|=n, and assume that there is no hyperplane containing both a translation ofA and a translation ofB. Under this condition it is proved that the number of distinct vectors in the form {a+b∶a∈A, b∈B} is at leastn+dm−d(d+1)/2. This generalizes results of Freiman (caseA=B) and Freiman, Heppes, Uhrin (caseA=−B). A more complicated estimate is also given which yields the exact bound for alln>2d.

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  1. [1]

    G. A. Freiman:Foundations of a structural theory of set addition (in Russian), Kazan Gos. Ped. Inst., Kazan 1966.

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  2. [2]

    G. A. Freiman:Foundations of a structural theory of set addition, Translation of Mathematical Monographs,37, Amer. Math. Soc., Providence, R. I., USA 1973.

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  3. [3]

    G. Freiman, A. Heppes, andB. Uhrin: A lower estimation for the cardinality of finite difference sets in ℝn,Proc. Conf. Number Theory, Budapest 1987, Coll. Math. Soc. J. Bolyai 51, North-Holland—Bolyai Társulat, Budapest (1989), 125–139.

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Supported by Hungarian National Foundation for Scientific Research, Grant No. 1901.

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Ruzsa, I.Z. Sum of sets in several dimensions. Combinatorica 14, 485–490 (1994).

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AMS subject classification code (1991)

  • 11 P 99
  • 11 B 75
  • 52 C 10