Higher-dimensional analog of the one-dimensional decomposition of almost periodic multisets

Determine whether, for any dimension d > 1, every almost periodic multiset A = {a_n}_{n∈N} ⊂ R^d with density D > 0 admits, under an appropriate numbering, a decomposition a_n = b_n + φ(n), where {b_n} enumerates the scaled cubic lattice D^{-1/d} Z^d and φ: N → R^d is an almost periodic mapping.

Background

In one dimension, it is known that every almost periodic multiset A ⊂ R, under appropriate numbering, can be written in the form a_n = n/D + φ(n), where D is the density of A and φ: N → R is an almost periodic mapping. This yields a canonical structural decomposition into a lattice-like part plus an almost periodic perturbation.

This paper proves that roughly shift-invariant sets (and hence almost periodic sets) in R^d are uniformly spread and can be matched to a scaled lattice with bounded displacement. The authors ask whether the one-dimensional structural representation has a precise analog in higher dimensions, i.e., whether a comparable decomposition exists for almost periodic multisets in R^d.