Sumsets in difference sets

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, $$\bar d$$(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 $$\bar d$$(B) > 0 and$$E - E \supset \bigcup\limits_{i = 1}^m {(p_i (B) \times q_i (B))} .$$(ii)  If A ⊂ ℤ with $$\bar d$$(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 E ⊂ ℤ3 with $$\bar d$$(E) > 0 such that E − E does not contain a set of the form B × B × B, where B ⊂ ℤ and $$\bar d$$(B) > 0.