### Abstract

It is shown that if A and B are non-empty subsets of {0, 1}^{ n} (for some nεN) then |A+B|≧(|A||B|)^{α} where α=(1/2) log_{2} 3 here and in what follows. In particular if |A|=2^{ n-1} then |A+A|≧3^{ n-1} which anwers a question of Brown and Moran. It is also shown that if |A| = 2^{ n-1} then |A+A|=3^{ n-1} if and only if the points of A lie on a hyperplane in n-dimensions. Necessary and sufficient conditions are also given for |A +B|=(|A||B|)^{α}. The above results imply the following improvement of a result of Talagrand [7]: if X and Y are compact subsets of K (the Cantor set) with m(X), m(Y)>0 then λ(X+Y)≧2(m(X)m(Y))^{α} where m is the usual measure on K and λ is Lebesgue measure. This also answers a question of Moran (in more precise terms) showing that m is not concentrated on any proper Raikov system.

Original language | English (US) |
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Pages (from-to) | 205-214 |

Number of pages | 10 |

Journal | Combinatorica |

Volume | 5 |

Issue number | 3 |

DOIs | |

State | Published - Sep 1 1985 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Computational Mathematics

### Keywords

- AMS subject classification (1980): 10E05

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## Cite this

*Combinatorica*,

*5*(3), 205-214. https://doi.org/10.1007/BF02579363