$ is $ B(1) $-lacunary. We claim that $ \mathcal{D} $ is on a rational grid in the sense that $ \rho(x,y) \in \mathbb{Q}^{d} $ for all $ x, y \in \mathcal{D} $ such that $ x \cber*[X] y $. Indeed, let $ m $ be so large that $ \phi_{0,\infty}^{-1}(x) \fber*[m][0] \phi_{0,\infty}^{-1}(y) $, and therefore also $$\phi_{m+1,\infty}^{-1}(x) \fber*[m][m+1] \phi_{m+1,\infty}^{-1}(y).$$ Item implies that $$\vec{q} := \rho\bigl(\phi_{m+1,\infty}^{-1}(x), \phi_{m+1,\infty}^{-1}(y)\bigr) \in
    \mathbb{Q}^{d}.$$ It now follows from item and the definition of the limit cross section that $ \rho(x,y) = \vec{q} $ and thus $ \rho(x,y) \in \mathbb{Q}^{d} $ as claimed.

The required rational grid is given by $ \mathcal{D} + \mathbb{Q}^{d} $.

Let $ Q $ be a rational grid for a Borel flow $ \mathbb{R}^{d} {\curvearrowright}X $. We say that a cross section $ \mathcal{C} $ is *on the grid* $ Q $ if $ \mathcal{C} \subseteq Q $. A small perturbation allows one to shift any cross section to a given grid.

\[lem:rational-perturbation\] Let $ Q \subseteq X $ be a rational grid for a free flow $ \mathbb{R}^{d} {\curvearrowright}X $. For any cross section $ \mathcal{C} \subseteq X $ and any $ \epsilon > 0 $ there exist a cross 