+ R_{i,n} \bigr) $ given by $$x \fber*[n][i] y \iff x, y \in (c + R_{i,n}) \quad \textrm{for some } c \in
    \mathcal{C}_{i,n}.$$ Each $ \fber*[n][i] $ class lives in a unique $ R_{i,n} $ box. We are going to construct trees of Borel partitions $ (\Omega_{i,s})_{s \in \mathbb{N}^{<\mathbb{N}}} $ on $ X_{i} $, together with families of positive integers $ (\omega_{i,s})_{s \in \mathbb{N}^{<\mathbb{N}}} $. Before listing properties of these objects, let us introduce the following sets: $$\begin{aligned}
      \mathcal{V}_{1,s} = \bigl\{ &x \in \mathcal{C}_{1, \omega_{1,s}} : \textrm{for all }
      y_{1},y_{2} \in \mathcal{D}_{1} \cap \bigl( x + R_{1,\omega_{1,s}}\bigr)
      \\ &\textrm{one has } \zeta(y_{1}) {\fber*[\omega_{2,s}][2]} \zeta(y_{2}) \bigr\},\\
      \mathcal{V}_{2,s} = \bigl\{ &x \in \mathcal{C}_{2, \omega_{2,s}} : \textrm{for all }
      y_{1},y_{2} \in \mathcal{D}_{2} \cap \bigl( x + R_{2,\omega_{2,s}}\bigr)
      \\ &\textrm{one has } \zeta^{-1}(y_{1}) {\fber*[\omega_{1,s}][1]} \zeta^{-1}(y_{2}) \bigr\}.\\
    \end{aligned}$$ Figure \[fig:sets-VXn\] illustrates the definition of the set $ \mathcal{V}_{1,s} $: a point $ x \in \mathcal{C}_{1,\omega_{1,s}}$ belongs to $ \mathcal{V}_{1,s} $ if the images under $ \zeta $ of all the points of $ \mathcal{D}_{1} $ in the box $ R_{1,\omega_{1,s}} $ around $ x $ fall into a single box $ R_{2,\omega_{2,s}} $ in $ X_{2} $. The definition of $ \mathcal{V}_{2,s} $ uses $ 