Minimal anisotropic assumptions for the Hausdorff-dimension lower bound

Identify minimal anisotropic assumptions on the spectral measure of a random countable stable zonotope that supplement uniform control of spherical belts with quantitative non-degeneracy of tangential components sufficient to establish the lower-bound argument for the Hausdorff dimension of its extreme-point set.

Background

The paper proves the exact Hausdorff dimension of the extreme points for uniformly distributed directions. The upper-bound argument only requires a uniform estimate on the spectral measure of thin spherical belts, whereas the lower bound relies on rotational invariance and a uniform two-sided estimate for the characteristic exponent of tangential field increments.

For general spectral measures, the canonical rotational reduction used in the proof is unavailable. The authors therefore leave unresolved the task of determining the weakest anisotropic conditions under which tangential increments remain quantitatively non-degenerate and the same dimension theory can be obtained.