Sumsets in difference sets

Vitaly Bergelson, Imre Z. Ruzsa

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

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.

Original languageEnglish
Pages (from-to)1-18
Number of pages18
JournalIsrael Journal of Mathematics
Volume174
Issue number1
DOIs
Publication statusPublished - Nov 2009

ASJC Scopus subject areas

  • Mathematics(all)

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