A realization theorem for sets of distances

Let H be an atomic monoid. The set of distances Δ(H) of H is the set of all d∈N with the following property: there are irreducible elements u1,…,uk,v1…,vk+d such that u1⋅…⋅uk=v1⋅…⋅vk+d but u1⋅…⋅uk cannot be written as a product of ℓ irreducible elements for any ℓ∈N with k<ℓ