A note on Diophantine approximation

We prove the existence of a dense subset Δ of [0,4] such that for all α∈Δ there exists a subgroup Xα of infinite rank of ℤ[z] such that Xα is a discrete subgroup of C[0,β] for all β≥α but it is not a discrete subgroup of C[0,β] for any β∈(0,α).