Besicovitch's example in higher dimensions: a purely unrectifiable set with large lower density

We generalize to arbitrary dimensions an example originally introduced by Besicovitch, obtaining for every $d \geq 1$ a purely $d$-unrectifiable set $E \subset \mathbb{R}^{d+1}$ such that $\Theta_{\ast}^{d}(E, x) = \liminf_{r \to 0} \mathcal{H}^{d}(E \cap B_{r}(x))/(2r)^{d} = 1/2$ for $\mathcal{H}^{d}$-almost every point $x \in E$. This establishes the lower bound $1/2$ for the minimal value $\sigma$ such that, if $\Theta_{\ast}^{d}(E, x) > \sigma$ for $\mathcal{H}^{d}$-almost all $x \in E$, then $E$ is $d$-rectifiable. This threshold was conjectured to be exactly $1/2$ by Besicovitch.