Optimal value of the perturbation constant in the uniformly spread bijection

Determine the minimal achievable bound for the constant C in Theorem 2 that guarantees, for every roughly shift-invariant discrete set A ⊂ R^d of density D, the existence of a bijection Θ from A onto the scaled lattice D^{-1/d} Z^d satisfying sup_{a∈A} |a − Θ(a)| ≤ C.

Background

Theorem 2 establishes that any roughly shift-invariant set A ⊂ R^d is uniformly spread: there exists a bijection from A onto D^{-1/d} Z^d with uniformly bounded displacement. The constructive proof produces an explicit, but not necessarily optimal, bound on the displacement constant C.

The authors ask for the optimal value of this perturbation constant C, i.e., the smallest bound that can be guaranteed under the rough shift-invariance assumptions. This problem concerns quantitative optimization of lattice approximation and displacement for uniformly spread sets.