section $ \mathcal{C}' $ on the grid $ Q $ and a Borel orbit equivalence $ \phi : \mathcal{C} \to \mathcal{C}' $ such that $ \bigl|\bigl|\rho\bigl(x, \phi(x)\bigr)\bigr|\bigr| < \epsilon $ for all $ x \in \mathcal{C} $.

Let $ \delta > 0 $ be so small that $ \mathcal{C} $ is $ B(\delta) $-lacunary. We may assume without loss of generality that $ \epsilon < \delta $. Let $ P \subseteq \mathcal{C} \times Q $ be the set $$P = \bigl\{ (x, y) \in \mathcal{C} \times Q : x \cber*[X] y \textrm{ and } ||\rho(x,y)|| <
    \epsilon \bigr\}.$$ Clearly, $ \operatorname{proj}_{\mathcal{C}}(P) = \mathcal{C} $. Since the projection of $ P $ onto the first coordinate is countable-to-one, Luzin–Novikov Theorem (see [@MR1321597 18.14]) guarantees existence of a Borel “inverse”, i.e., a Borel map $ \phi : \mathcal{C} \to Q $ such that $ \bigl(c, \phi(c)\bigr) \in P $ for all $ c \in \mathcal{C} $. The map $ \phi $ is injective, since $ \epsilon < \delta $. The required cross section $ \mathcal{C}' $ is given by $ \phi(\mathcal{C}) $.

Some Simple Tools {#sec:some-simple-tools}
=================

In this section we gather a few elementary tools that will be useful in the 