Multiplicity, rank pairs (Q1361068)

This paper is the culmination of the work of numerous authors concerning the problem of M. K. Mentzen. Given a dynamical system \((X,\mu,T)\), the rank \(r(T)\) and spectral multiplicity \(m(T)\) satisfy \(1\leq m(T)\leq r(T)\leq\infty\). \textit{M. K. Mentzen} [Bull. Pol. Acad. Sci., Math. 35, 417-424 (1987; Zbl 0675.28006)] conjectured that any such pair of integers (or \(\infty\)) is attainable by some \textit{ergodic} dynamical system on a non-atomic Lebesgue probability space. Contributions to the solution of this problem have been made by R. V. Chacon, A. del Junco, A. E. Robinson, G. R. Goodson, M. Lemańczyk, M. K. Mentzen, S. Ferenczi, J. Kwiatkowski and C. Mauduit. This paper completes this line of research by answering Mentzen's question affirmatively.