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 $d\ge 1$ a purely $d$-unrectifiable set $E\subset {\mathbb{R}}^{d+1}$ such that ${\mathrm{\Theta }}_{\ast }^d(E,x)={\mathrm{lim\hspace{0.17em}inf}}_{r\to 0}{\mathcal{H}}^d(E\cap B_r(x))\mathrm{/}(2r{)}^d=1\mathrm{/}2$ for ${\mathcal{H}}^d$-almost every point $x\in E$. This establishes the lower bound $1\mathrm{/}2$ for the minimal value $\sigma$ such that, if ${\mathrm{\Theta }}_{\ast }^d(E,x)>\sigma$ for ${\mathcal{H}}^d$-almost all $x\in E$, then $E$ is $d$-rectifiable. This threshold was conjectured to be exactly $1\mathrm{/}2$ by Besicovitch.