### 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 - Jan 2010 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Israel Journal of Mathematics*,

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

**Sumsets in difference sets.** / Bergelson, Vitaly; Ruzsa, I.

Research output: Contribution to journal › Article

*Israel Journal of Mathematics*, vol. 174, no. 1, pp. 1-18. https://doi.org/10.1007/s11856-009-0100-3

}

TY - JOUR

T1 - Sumsets in difference sets

AU - Bergelson, Vitaly

AU - Ruzsa, I.

PY - 2010/1

Y1 - 2010/1

N2 - We study some properties of sets of differences of dense sets in ℤ2 and ℤ3 and their interplay with Bohr neighbourhoods in ℤ. We obtain, inter alia, the following results. (i) If E ⊂ ℤ2, d̄(E) > 0 and pi, qi ∈ ℤ[x], i = 1, ..., m satisfy pi(0) = qi(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 ⊂ ℤ3 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.

AB - We study some properties of sets of differences of dense sets in ℤ2 and ℤ3 and their interplay with Bohr neighbourhoods in ℤ. We obtain, inter alia, the following results. (i) If E ⊂ ℤ2, d̄(E) > 0 and pi, qi ∈ ℤ[x], i = 1, ..., m satisfy pi(0) = qi(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 ⊂ ℤ3 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.

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U2 - 10.1007/s11856-009-0100-3

DO - 10.1007/s11856-009-0100-3

M3 - Article

VL - 174

SP - 1

EP - 18

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

IS - 1

ER -