which is also called a $k$*-cube*. Note that we think of the $Q_I$ as being embedded in the torus $\T_n$ it is possible that $Q_I\cap Q_{I^\p}\ne\es$, even if the $\d_\infty$-distance between $z_I$ and $z_{I^\p}$ is strictly larger than $a_k$. We say that $I,I^\p\in J_k$ are $*$-connected if $I_1=I,\ldots,I_m=I^\p$ $Q_{I_j}\cap Q_{I_{j+1}}\ne\es$ $j\in\{1,\ldots,m-1\}$. In the following, we frequently consider certain neighborhoods of cubes of the form $Q_I$ for some $I\in J_k$. To be more precise, for $r>0$ we denote by $Q_I^r=\{\xi\in\T_n: \d^{\T_n}_\infty(\xi,Q_I)\le r\}$ the subset of all $\xi\in\T_n$ the toroidal $\d_\infty$-distance from $\xi$ to the cube $Q_I$ is at most $r$.

Next, we need to capture the property that a path $\gamma$ in $G(X^{(n)})$ that starts in a cube $Q_I$ needs a large number of hops to move far away from this cube. To be more precise, for $\varepsilon>0$ and $k\ge1$ we introduce the notion of $(\varepsilon,k)$-good indices. If $I\in J_k$, then $I$ is always $(\varepsilon,k)$-good. , inductively, if $I\in J_{k^\p}$ is $0\le k^\p<k$, then we say that $I$ is $(\varepsilon,k)$-good if a) $X^{(n)}\cap (Q_I\times[a_{k^\p+1},a_{k^\p}])=\es$ and b) for every $*$-connected subset $\gamma\subset J_{k^\p+1}$ that is contained in $Q^{a_{k^\p}}_I$ and consists of at least $b_{k^\p+1}/4$ elements, it holds that $\gamma$ contains at most 