Unitary operators with discrete spectrum induced by measure preserving transformations

The purpose of this paper is to prove the following theorem.Theorem. Given a countable subset Λ on the circle K and an integer-valued function n(λ) on Λ, there exists a dynamical system with discrete spectrum (X,$\text{F}$,μ,T) such that Λ is the set of all eigenvalues of T and n(λ) is the multiplicity function of T if and only if there exist two systems of subgroups {G_i}_i∈N and $\text{{S}{}_{\mathrm{j}}\text{}}{}_{\mathrm{j\in M}}\text{(}\text{N}\text{??}{}_0\text{,}\text{M}\text{??}{}_0\text{)}$ of the circle such that Λ = G ∪ S and

$\text{n(λ)=}\text{∞}\text{for}\text{λ∈S,}\text{{i∈N; λ∈G}{}_{\mathrm{i}}\text{}}\text{for}\text{λ∈G?S,}$

where $\text{G =}\text{?}\text{i∈N}\text{G}{}_{\mathrm{i}}\text{, S =}\text{?}\text{j∈M}\text{S}{}_{\mathrm{j}}$.