Determine the covering number of generic half-grids in arbitrary dimension

Determine the covering number for generic d-dimensional half-grids, proving whether it equals (1 + 1/2 + ⋯ + 1/(d+1) + 1/(d+2))nk − O(n^{d−1} + k).

Background

The paper establishes asymptotically sharp multiplicity-covering results for generic half-grids in dimensions two and three. In dimension two, the covering number is approximately (3/2)nk, while in dimension three it is approximately (11/6)nk. Based on these results, the authors propose a general formula for generic half-grids in dimension d, with an error term of order O(n{d−1} + k). The statement is presented explicitly as a conjecture and is not proved in the paper.

References

Inspired by our results in dimensions 2 and 3, we conjecture the following general bound. Conjecture 1. For generic d-dimensional half-grids, the covering number is equal to (1 + 2 + ... + d+I + 2)nk - O(nd-1 + k).

— Covering half-grids with lines and planes  (2501.11156 - Bishnoi et al., 19 Jan 2025) in Section 4, Conclusion