Lemma 3.24 of Additive Combinatorics by Tao and Vu states the following:
Let $\Gamma \subset \mathbb{R}^d$ be a lattice of full rank, let $V$ be a bounded open subset of $\mathbb{R}^d$, and let $P$ be a finite non-empty set in $\mathbb{R}^d$. Then, $$ |(V-V) \cap (\Gamma + P-P) \geq \dfrac{\text{mes}(V) |P|}{\text{mes}(\mathbb{R}^d/\Gamma)}$$
I have a doubt what happens in the following scenario: take $\Gamma = \mathbb{Z}^d$ and $P$ to be an arbitrarily large subset of $\mathbb{Z}^d$. Then the LHS is $|(V-V) \cap \Gamma|$ which does not depend on $P$, whereas the RHS is increasing with $|P|$. Why doesn't this give a contradiction when $|P|$ is sufficiently large?