h_{n} : \mathcal{C}_{n} \to B(a_{i}) $, then the spiral $ \mathfrak{C} $ is convergent. If furthermore $ \mathcal{C}_{0} $ is $ B(\delta) $-lacunary for some $ \delta $ such that $ \sum_{i}a_{i} < \delta $, then the limit cross section of the spiral is $ B\bigl(\delta - \sum_{i}a_{i}\bigr) $-lacunary.

The proof is immediate from the equation .

A *rational grid* for a flow $ \mathbb{R}^{d} {\curvearrowright}X $ is a Borel subset $ Y \subseteq X
   $ which is invariant under the action of $ \mathbb{Q}^{d} $ and intersects every orbit of the flow in a unique $ \mathbb{Q}^{d} $-orbit: for every $ x \in X $ there is $ y \in Y $ such that $
   [x]_{\mathbb{R}^{d}} \cap Y = [y]_{\mathbb{Q}^{d}} $.

\[lem:existence-of-rational-grids\] Any free Borel flow admits a rational grid.

For every $ \epsilon > 0 $ we pick a Borel map $ \alpha_{\epsilon} : \mathbb{R}^{d} \to \mathbb{Q}^{d} $ such that $ \bigl|\bigl| \vec{r} - \alpha_{\epsilon}(\vec{r}) \bigr|\bigr| < \epsilon $ for all $ \vec{r} \in \mathbb{R}^{d} $. By a theorem of Kechris [@MR1176624], there exists a $ B(2) $-lacunary cross section $ \mathcal{C} \subseteq X $. Restriction $ \cber[\mathcal{C}] $ of the orbit equivalence relation onto $ \mathcal{C} $ 