Universal Ahlfors--David regularity of Steiner trees

The celebrated Steiner tree problem is the problem of finding a set $St$ of minimum one-dimensional Hausdorff measure $H$ (length) such that $St \cup \mathcal{A}$ is connected, where $\mathcal{A} \subset \mathbb{R}^d$ is a given compact set. Paolini and Stepanov provided very general existence and regularity results for the Steiner problem. Their main regularity result is that under a natural assumption, $H(St) < \infty$, for almost every $\varepsilon>0$ the set $St_\varepsilon := St\setminus B_\varepsilon(\mathcal A)$ is an embedded finite forest (acyclic graph). We give a quantitative regularity result by proving that the set $St_\varepsilon$ is Ahlfors--David regular with constants that depend only on $d$ (and not on $\mathcal{A}$). Namely, for $d > 2$, every $\varepsilon > 0$, every $x \in St_\varepsilon$, and every choice of $\rho \in (0,1)$, we have $$\frac{H \left (St_\varepsilon \cap B_{\rho \varepsilon}(x) \right) }{\varepsilon} \leq \left ( \frac{144d}{1-\rho} \right) ^{d-2}.$$ As a corollary, we obtain a density-type result, i.e. that the set $St_\varepsilon \cap B_{\rho \varepsilon}(x)$ consists of at most $$\left ( \frac{144d}{1-\rho} \right) ^{d-1}$$ line segments. In the plane (i.e., for $d=2$), it is possible to obtain tight structural results.