Research proof index

Consider two tiles $p_1,p_2$ with $\mathfrak s(p_1)\le\mathfrak s(p_2)$ whose enlarged spatial balls overlap. If $x_i$ lies in the active set $E(p_i)$, then the separation of the two tile frequencies, measured at the scale of $p_1$, is controlled by the separation of the selected phases $Q(x_1)$ and $Q(x_2)$:

$1+d_{p_1}(\mathcal Q(p_1),\mathcal Q(p_2))\le C(a)\bigl(1+d_{x_1,D^{\mathfrak s(p_1)}}(Q(x_1),Q(x_2))\bigr).$