\mathcal{C}_{k} \to \mathbb{R}^{d} $ by $$H_{k}(x) = \lim_{n \to \infty} \Bigl[ h_{k}(x) + h_{k+1}\bigl(\phi_{k,k+1}(x)\bigr) + \cdots +
   h_{n-2}\bigl(\phi_{k,n-2}(x)\bigr) + h_{n-1}\bigl(\phi_{k,n-1}(x)\bigr)\Bigr].$$ Being a pointwise limit of Borel functions, $ H_{k} $ is Borel. The *limit cross section* of a convergent spiral is a set $ \mathcal{D} \subseteq X $ defined by $$\mathcal{D} = \mathcal{C}_{0} + H_{0} = \{ x + H_{0}(x) : x \in \mathcal{C}_{0} \}.$$ Note that $ \mathcal{D} = \mathcal{C}_{k} + H_{k} $ for any $ k \in \mathbb{N} $. Also, we let $ \phi_{k, \infty} : \mathcal{C}_{k} \to \mathcal{D} $ to be $ \phi_{k,\infty}(x) = x + H_{k}(x) $. The set $ \mathcal{D} $ is necessarily Borel, as it is a countable-to-one image of a Borel function. In general, $ \mathcal{D} $ may not be lacunary, but the following easy conditions guarantee lacunarity of the limit cross section. Hereafter $ B(\delta) \subseteq \mathbb{R}^{d} $ denotes an open ball of radius $ \delta $ around the origin.

\[prop:lacunarity-of-limit-cross-section\] Let $ \mathfrak{C} = (\mathcal{C}_{n}, h_{n}) $, $ n \in \mathbb{N} $, be a spiral of cross sections for a free flow $ \mathbb{R}^{d} {\curvearrowright}X $. If there exists a convergent series $ \sum_{i=0}^{\infty}a_{i} $ of positive reals such that $ 