# Sum of sets in several dimensions

Research output: Contribution to journalArticle

22 Citations (Scopus)

### Abstract

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 language English 485-490 6 Combinatorica 14 4 https://doi.org/10.1007/BF01302969 Published - Dec 1994

Hyperplane
Finite Set
Distinct
Generalise
Estimate
Form

### Keywords

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

### ASJC Scopus subject areas

• Discrete Mathematics and Combinatorics
• Mathematics(all)

### Cite this

In: Combinatorica, Vol. 14, No. 4, 12.1994, p. 485-490.

Research output: Contribution to journalArticle

Ruzsa, I. / Sum of sets in several dimensions. In: Combinatorica. 1994 ; Vol. 14, No. 4. pp. 485-490.
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