Sum of sets in several dimensions

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

Original languageEnglish
Pages (from-to)485-490
Number of pages6
Issue number4
Publication statusPublished - Dec 1 1994



  • AMS subject classification code (1991): 11P99, 11B75, 52C10

ASJC Scopus subject areas

  • Discrete Mathematics and Combinatorics
  • Computational Mathematics

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