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A density theorem for (multiplicity, rank) pairs. (English) Zbl 0833.28010
Recall that associated with a dynamical system \((X,T, \mu)\) there are certain invariants. The rank \(r\), a conjugacy invariant, is the number of Rokhlin towers necessary to generate the system. The (spectral) multiplicity \(m\), a spectral invariant, is the number of cyclic subspaces necessary to span the \(L^2\)-space for the associated \(L^2\)- operator. It is known that for a dynamical system \(m \leq r\) (Chacon). M. K. Mentzen [Bull. Pol. Acad. Sci., Math. 35, 417-424 (1987; Zbl 0675.28006)] conjectured that for each pair \((m,r)\) with \(m \leq r\) (natural numbers), it should be possible to construct a dynamical system with multiplicity \(m\) and rank \(r\), and he showed this is possible for pairs of the form \((1,n)\). Subsequently, this reviewer and M. Lemańczyk [Stud. Math. 96, No. 3, 219-230 (1990; Zbl 0711.28007)] showed that \((2,n)\) can be achieved. More recently, other authors have constructed pairs of the form \((n,n)\), \((n,2n)\) and \((p,p - 1)\), \(p\) prime.
In this paper the authors construct a new class of examples giving rise to pairs of the form \((d,n)\) for all \(n \geq 3\) and all \(d\) dividing an arithmetic function \(\psi (n)\). In particular, it follows that for fixed \(d\), the set of possible \(n\) is of density one in \({\mathbf N}\).

MSC:
28D10 One-parameter continuous families of measure-preserving transformations
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