Sharpness of Klartag’s lattice-packing bound

Determine whether Klartag’s asymptotic lower bound of order d^2/2^d is sharp up to constant factors for lattice sphere packings.

Background

The paper describes Klartag’s improvement of the high-dimensional sphere-packing lower bound to a quantity of order d2/2d for lattice packings. It identifies the matching upper bound, up to constant factors, as an unresolved issue specific to lattice packings.

References

Interestingly, it seems reasonable to conjecture that Klartag's bound is also sharp, up to constant factors, in the case of lattice packings.

— Probabilistic combinatorics at exponentially small scales  (2512.15077 - Sahasrabudhe, 17 Dec 2025) in Section 1, Introduction; discussion of sphere packing in high dimensions