### Abstract

We study some properties of sets of differences of dense sets in ℤ^{2} and ℤ^{3} and their interplay with Bohr neighbourhoods in ℤ. We obtain, inter alia, the following results. (i) If E ⊂ ℤ^{2}, d̄(E) > 0 and p_{i}, q_{i} ∈ ℤ[x], i = 1, ..., m satisfy p_{i}(0) = q_{i}(0) = 0, then there exists B ⊂ ℤ such that d̄(B) > 0 and, (ii) If A ⊂ ℤ with d̄(A) > 0, then for any r, s, t such that r + s + t = 0 the set rA + sA + tA is a Bohr neighbourhood of 0. (iii) For any 0 < α < 1/2 there exists a set E ⊂ ℤ^{3} with d̄(E) > 0 such that E - E does not contain a set of the form B × B × B, where B ⊂ ℤ and d̄(B) > 0.

Original language | English |
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Pages (from-to) | 1-18 |

Number of pages | 18 |

Journal | Israel Journal of Mathematics |

Volume | 174 |

Issue number | 1 |

DOIs | |

Publication status | Published - Nov 2009 |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Bergelson, V., & Ruzsa, I. Z. (2009). Sumsets in difference sets.

*Israel Journal of Mathematics*,*174*(1), 1-18. https://doi.org/10.1007/s11856-009-0100-3