### Abstract

Let A, B be finite sets in ℝ^{d} 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 |
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Pages (from-to) | 485-490 |

Number of pages | 6 |

Journal | Combinatorica |

Volume | 14 |

Issue number | 4 |

DOIs | |

Publication status | Published - Dec 1994 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Mathematics(all)

### Cite this

*Combinatorica*,

*14*(4), 485-490. https://doi.org/10.1007/BF01302969

**Sum of sets in several dimensions.** / Ruzsa, I.

Research output: Contribution to journal › Article

*Combinatorica*, vol. 14, no. 4, pp. 485-490. https://doi.org/10.1007/BF01302969

}

TY - JOUR

T1 - Sum of sets in several dimensions

AU - Ruzsa, I.

PY - 1994/12

Y1 - 1994/12

N2 - 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.

AB - 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.

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

UR - http://www.scopus.com/inward/record.url?scp=0001040462&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001040462&partnerID=8YFLogxK

U2 - 10.1007/BF01302969

DO - 10.1007/BF01302969

M3 - Article

AN - SCOPUS:0001040462

VL - 14

SP - 485

EP - 490

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 4

ER -