Let $G$ be a countable amenable group with a (left) Følner sequence $(F_n)$. Let $\Gamma$ be a subset of $G$ that has density $1$ with respect to $(F_n)$, in the sense that
$$
\lim_{n \to \infty} \frac{|\Gamma \cap F_n|}{|F_n|} \ = \ 1.
$$$$
\lim_{n \to \infty} \frac{\lvert\Gamma \cap F_n\rvert}{\lvert F_n\rvert} \ = \ 1.
$$
Now define
$$
\Gamma' := \{ (s,t) \in G^2 : st^{-1} \in \Gamma \}.
$$
Is it true that $\Gamma'$ has density $1$ in $G^2$ with respect to $(F_n \times F_n)$?
A starting point is the calculation $$ |\Gamma' \cap F_n^2| = \sum_{\gamma \in \Gamma} \#\{(s,t) \in F_n^2 : st^{-1} = \gamma \} \ = \ \sum_{\gamma \in \Gamma} |F_n \cap \gamma^{-1} F_n|. $$$$ \lvert\Gamma' \cap F_n^2\rvert = \sum_{\gamma \in \Gamma} \#\{(s,t) \in F_n^2 : st^{-1} = \gamma \} \ = \ \sum_{\gamma \in \Gamma} \lvert F_n \cap \gamma^{-1} F_n\rvert. $$
In the special case $G = \mathbb{Z}$, $F_n = [-n,n]$, the result is true, but my argument uses the geometry of $\mathbb{Z}$ quite strongly. The argument is: in this case, $|F_n \cap \gamma^{-1}F_n| = \max(2n+1 - |\gamma|, 0)$$\lvert F_n \cap \gamma^{-1}F_n\rvert = \max(2n+1 - \lvert\gamma\rvert, 0)$, so if $n$ is chosen large enough so that $|\Gamma \cap F_n| > (1-\epsilon)|F_n|$$\lvert\Gamma \cap F_n\rvert > (1-\epsilon)|F_n|$, then $$ \sum_{\gamma \in \Gamma} |F_n \cap \gamma^{-1} F_n| \ \geq \ \sum_{\epsilon \cdot n < |\gamma| \leq 2n} (2n+1-|\gamma|) \ \geq \ (2n+1)^2 \cdot (1 - O(\epsilon)) \ = \ |F_n^2|(1-O(\epsilon)) $$$$ \sum_{\gamma \in \Gamma} \lvert F_n \cap \gamma^{-1} F_n\rvert \ \geq \ \sum_{\epsilon \cdot n < \lvert\gamma\rvert \leq 2n} (2n+1-\lvert\gamma\rvert) \ \geq \ (2n+1)^2 \cdot (1 - O(\epsilon)) \ = \ \lvert F_n^2\rvert(1-O(\epsilon)) $$ as desired.
Is this result true for all countable amenable groups?
