Distance sets bounds for polyhedral norms via effective dimension

We prove that, for every norm on \$\mathbb{R}^d\$ and every \$E \subseteq \mathbb{R}^d\$, the Hausdorff dimension of the distance set of \$E\$ with respect to that norm is at least \$\dim_{\mathrm{ H}} E - (d-1)\$. An explicit construction follows, demonstrating that this bound is sharp for every polyhedral norm on \$\mathbb{R}^d\$. The techniques of algorithmic complexity theory underlie both the computations and the construction.

Joint work with Iqra Altaf.