$X$ is a Cantor set, then the assumption holds for any $\theta>0$. Normality implies the existence of $\theta$-Hölder partitions of unity: see e.g. [@Katz p. 221].

Hölder vector bundles
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Let ${\mathbb{E}}$ be a $d$-dimensional vector bundle over $X$. We recall the definition and fix the terminology. ${\mathbb{E}}$ is a topological space endowed with a continuous map ${\pi \colon {\mathbb{E}}\to X}$ (called the *projection*), a cover of $X$ by open sets $U_i$ (called *coordinate neighborhoods*), and a family of homeomorphisms (called *coordinate maps*) $$\psi_i \colon U_i \times {\mathbb{R}}^d \to \pi^{-1}(U_i) \quad
\text{such that $\pi( \psi_i(x, u) ) = x$ for all $(x,u) \in  U_i \times {\mathbb{R}}^d$,}$$ which is required to have the following compatibility property: whenever $x \in U_i \cap U_j$, the map $$g_{j \gets i}(x) \coloneqq \big[ \psi_j(x, \mathord{\cdot})\big]^{-1} \circ \psi_i(x, \mathord{\cdot}) \colon {\mathbb{R}}^d \to {\mathbb{R}}^d$$ is linear. Therefore we obtain a family of continuous maps: $$\label{e.g}
g_{j \gets i} \colon U_i \cap U_j \to {\mathit{GL}}(d,{\mathbb{R}}),$$ which are called *coordinate transformations*. Moreover, each *fiber* ${\mathbb{E}}_x \coloneqq \pi^{-1}(x)$ has a unique structure of $d$-dimensional vector space such that the maps $$\label{e.h_i}
h_i(x) \coloneqq \psi_i(x, \mathord{\cdot}) \colon {\mathbb{R}}^d \to {\mathbb{E}}_x$$ become isomorphisms. Since $X$ is assumed to be compact, we will 