In the original paper of Ornstein and Weiss "entropy and isomorphism theorems for actions of amenable groups" I have trouble understanding the proof of the quasi-tiling lemma for countable groups. In step (2) one wants to cover a set $D$ that is invariant with respect to the largest tile $T_N$ meaning that the $T_NT_N^{-1}$-boundary of $D$ is smaller than $\delta$ times the size of $D$.
More precisely, $A$ is called $(K,\delta)$-invariant if $|\partial_K A|=|\{g\in G: Kg\cap A\ne\emptyset\ne Kg\cap A^c\}|<\delta |A|$. Note that $\{g\in A:Kg\subseteq A\}=A\setminus \partial_K A$, so around every point in $A$ that is at least "$K$-away" from the boundary, one can put $K$ and it will still be contained in $A$.
Now using the invariance of $D$, it can be tiled by translates of $T_N$ that cover a more than $(1-\delta)$ of $D$. I would like to leave out the details of the previous lemmas, because the problem I have is just with the next step. They proceed to select some of the covering translates of $T_N$ that are "almost disjoint" and still cover $\epsilon(1-\delta)$ and then take out the already covered part $D_1=D\setminus T_NC_N$ and claim that $(1-\delta_1)$ of $D_1$ can be covered by translates of $T_{N-1}$, where $\delta_1<2\delta+2\eta_{N-1}$. Here $T_{N-1}$ is the next smallest tile which by the invariance assumptions is smaller than roughly a constant times $\eta_{N-1}$ of $T_N$.
I can't figure out how to prove this claim. One way would certainly be to show that also $D_1$ is $(T_{N-1}T_{N-1}^{-1},\delta_1)$-invariant. But I can't prove the required inequality. We can calculate that $|\partial_{T_{N-1}T_{N-1}^{-1}}D_1|\le |\partial_{T_NT_N^{-1}}D|+|\partial_{T_{N-1}T_{N-1}^{-1}} T_NC_N|\le \delta |D|+\eta_{N-1}|T_N||C_N|$ but this appears to be already too large.
If you read this paper or know any alternative sources where the original paper has been "worked up" a touch, I would greatly appreciate you help.