Necessity of averaging over radii in discrepancy lower bounds

Determine whether averaging over the spherical cap height or ball radius is necessary to obtain large discrepancy lower bounds for point distributions, specifically whether one can establish large discrepancy estimates without averaging by using a single fixed radius for spherical caps on the unit sphere S^d or balls on compact two-point homogeneous spaces.

Background

Classical lower bounds for spherical cap discrepancy on the sphere (Beck) and for ball discrepancy on compact two-point homogeneous spaces (Skriganov) are obtained by averaging over the cap height or radius. Subsequent work (e.g., Montgomery on the torus) shows that averaging over a small finite set of radii or shapes can suffice in some settings, and recent results (including those in this paper) identify specific single radii and dimensional conditions under which sharp or near-sharp single-radius bounds can be proven.

Despite these advances, the fundamental question remains whether such averaging is inherently required to obtain large discrepancy estimates across general spaces or whether single-radius bounds can be established universally. This uncertainty is explicitly acknowledged by the authors in the introduction.