Sumsets in difference sets

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, $\bar d$(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 < 1/2 there exists a set 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.