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6. Structure in$~(A+B)\cdot (C+D)$

Theorem.
(Fish [arXiv:1702.02544]) If $A,B\subseteq\mathbb{Z}$ have positive Banach density then $(A-A)\cdot (B-B)$ contains a nontivial subgroup of $\mathbb{Z}$. Moreover, the index of the subgroup depends quantitatively on the densities of $A$ and $B$.

Theorem.
(Björklund and Fish [MR3483067]) If $E\subseteq\mathbb{Z}^3$ has positive Banach density then $\{xy-z^2:(x,y,z)\in E-E\}$ contains a nontrivial subgroup of $\mathbb{Z}$.
1. Problem 6.1.

[Alexander Fish] Suppose $A\subseteq\mathbb{N}\times\mathbb{N}$ has positive Banach density, and set $\Delta=\{x\cdot y:(x,y)\in A-A\}$. Does $\Delta$ contain a nontrivial subgroup of $\mathbb{Z}$?
• Problem 6.2.

[Alexander Fish] Find a quantitative version of the second theorem above (i.e. express the index of the subgroup in terms of the Banach density of $A$).

Cite this as: AimPL: Nonstandard methods in combinatorial number theory, available at http://aimpl.org/nscombinatorial.