We study the structure of the difference set $E - E$ for subsets $E \subseteq \mathbb {Z}^2$ 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 $k \cdot \mathbb {Z} \subseteq \{ xy : (x,y) \in E - E \}$. Although this question remains open, we establish a relatively weaker form. Specifically, we prove that if $\langle a_j\rangle _{j=1}^m$ is any finite sequence in $\mathbb {N},$ then there exist infinitely many integers $k \in \mathbb {Z}$ and infinitely many sequences $\langle x_n \rangle _{n \in \mathbb {N}}$ in $\mathbb {Z}$ such that $k \cdot MT(\langle a_j \rangle _{j=1}^m, \langle x_n\rangle _{n}) \subseteq \{ xy : (x,y) \in E - E \}$, where $MT(\langle a_j \rangle _{j=1}^m, \langle x_n\rangle _{n})$ denotes the Milliken–Taylor configuration generated by the sequences $\langle a_j\rangle _{j=1}^m$ and $\langle x_n \rangle _{n \in \mathbb {N}}$.

For $E \subset \mathbb {N}$, a subset $R \subset \mathbb {N}$ is E-intersective if for every $A \subset E$ having positive relative density, $R \cap (A - A) \neq \varnothing $. We say that R is chromatically E-intersective if for every finite partition $E=\bigcup _{i=1}^k E_i$, there exists i such that $R\cap (E_i-E_i)\neq \varnothing $. When $E=\mathbb {N}$, we recover the usual notions of intersectivity and chromatic intersectivity. We investigate to what extent the known intersectivity results hold in the relative setting when $E = \mathbb {P}$, the set of primes, or other sparse subsets of $\mathbb {N}$. Among other things, we prove the following: (1) the set of shifted Chen primes $\mathbb {P}_{\mathrm {Chen}} + 1$ is both intersective and $\mathbb {P}$-intersective; (2) there exists an intersective set that is not $\mathbb {P}$-intersective; (3) every $\mathbb {P}$-intersective set is intersective; (4) there exists a chromatically $\mathbb {P}$-intersective set which is not intersective (and therefore not $\mathbb {P}$-intersective).

We show that if $\mathcal {A}\subset \{1,\ldots,N\}$ has no solutions to $a-b=n^2$ with $a,b\in \mathcal {A}$ and $n\geq 1$, then

\[ \lvert \mathcal{A}\rvert \ll \frac{N}{(\log N)^{c\log\log \log N}} \]

$c>0$

We demonstrate that every difference set in a finite Abelian group is equivalent to a certain ‘regular’ covering of the lattice $ A_n = \{ \boldsymbol {x} \in \mathbb {Z} ^{n+1} : \sum _{i} x_i = 0 \} $ with balls of radius $ 2 $ under the $ \ell _1 $ metric (or, equivalently, a covering of the integer lattice $ \mathbb {Z} ^n $ with balls of radius $ 1 $ under a slightly different metric). For planar difference sets, the covering is also a packing, and therefore a tiling, of $ A_n $. This observation leads to a geometric reformulation of the prime power conjecture and of other statements involving Abelian difference sets.