ON DIFFERENCE SETS OF DENSE SUBSETS OF Z 2 $\mathbb {Z}^2$ double struck upper Z squared

Abstract

Abstract We study the structure of the difference set E − E $E - E$ upper E minus upper E for subsets E ⊆ Z 2 $E \subseteq \mathbb {Z}^2$ upper E subset of or equal to double struck upper Z squared of positive upper Banach density. Fish [‘On product of difference sets for sets of positive density’, Proc. Amer. Math. Soc. 146 (2018), 3449–3453] asked in Problem 2 whether, for every such set E , there exists a nonzero integer k such that k ⋅ Z ⊆ { x y : ( x , y ) ∈ E − E } $k \cdot \mathbb {Z} \subseteq \{ xy : (x,y) \in E - E \}$ k dot double struck upper Z subset of or equal to StartSet x y colon left parenthesis x comma y right parenthesis element of upper E minus upper E EndSet . Although this question remains open, we establish a relatively weaker form. Specifically, we prove that if ⟨ a j ⟩ j = 1 m $\langle a_j\rangle _{j=1}^m$ left angle bracket a Subscript j Baseline right angle bracket Subscript j equals 1 Superscript m is any finite sequence in N , $\mathbb {N},$ double struck upper N comma then there exist infinitely many integers k ∈ Z $k \in \mathbb {Z}$ k element of double struck upper Z and infinitely many sequences