Generalize the generic half-grid covering number to arbitrary dimension

Establish the covering number for generic d-dimensional half-grids of order n when every point except the origin must be covered at least k times by hyperplanes avoiding the origin, with the asymptotic form conjectured by the authors.

Background

The paper determines asymptotically sharp covering numbers for generic half-grids in dimensions two and three. In dimension two, the relevant objects are lines covering all points except the vertex; in dimension three, they are planes covering all points except the origin. The conclusion proposes an analogous asymptotic formula for generic half-grids in every dimension, extending these low-dimensional results.