ON DIFFERENCE SETS OF DENSE SUBSETS OF Z 2 Z² double struck upper Z squared

We study the structure of the difference set E − E $E - E$ upper E minus upper E for subsets E ⊆ Z 2 E ⊆ Z² 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 · Z ⊆ \ xy : (x,y) ∈ 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 aⱼ ⱼ₌₁ᵐ left angle bracket a Subscript j Baseline right angle bracket Subscript j equals 1 Superscript m is any finite sequence in N , N, double struck upper N comma then there exist infinitely many integers k ∈ Z k ∈ Z k element of double struck upper Z and infinitely many sequences