Distance sets with respect to polyhedral norms

Let \$E\subseteq \mathbb{R}^d\$. The Falconer distance problem is to relate the dimension of  to the size of its distance set \$\Delta (E):=\{\|x-y\|:x,y\in E\}\$. After surveying the problem and some of the progress that has been made toward its solution, we consider the role of curvature and arrive at a bound on the dimension of pinned distance sets with respect to arbitrary norms—one that is sharp for all polyhedral norms. Its proof is grounded in effective methods, leading us to study Kolmogorov complexity and its surprising role in geometric measure theory. With the basic tools in hand, we prove the distance set estimate and sketch the ideas underlying its sharpness.