Covering number of generic half-grids in arbitrary dimension

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

Background

The paper proves asymptotically sharp covering results for generic half-grids in dimensions two and three. In the conclusion, the authors propose a general formula for generic half-grids in arbitrary dimension, extending the dimension-specific results and the harmonic-sum behavior observed for related full-grid covering problems.

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, Conjecture 1